Kṛṣṇa first appeared before His parents Devakī and Vasudeva in a four-handed form, and then He transformed Himself into the two-handed form. This mystery is very difficult to understand for those who are atheists or who are devoid of devotional service.
(Bhagavad-Gita----11:53----purport).

Bhagavān means the Almighty God who is the controller of all opulences, power, fame, beauty, knowledge and renunciation. He is the protector of His pure devotees. Although God is equally disposed to everyone, He is especially inclined to His devotees. Sat means the Absolute Truth. And persons who are servitors of the Absolute Truth are called sātvatas. And the Personality of Godhead who protects such pure devotees is known as the protector of the sātvatas. Bhadraṁ te, or "blessings upon you," indicates the sages' anxiety to know the Absolute Truth from the speaker. Lord Śrī Kṛṣṇa, the Supreme Personality of Godhead, appeared to Devakī, the wife of Vasudeva. Vasudeva is the symbol of the transcendental position wherein the appearance of the Supreme Lord takes place.
(Srimad Bhagavatam----1:1:12----purport).

Devakī, the mother of Kṛṣṇa and sister of King Kaṁsa, was put into prison along with her husband, Vasudeva, because the envious King was afraid of being killed by Devakī's eighth son (Kṛṣṇa). He killed all the sons of Devakī who were born before Kṛṣṇa, but Kṛṣṇa escaped the danger of child-slaughter because He was transferred to the house of Nanda Mahārāja, Lord Kṛṣṇa's foster father. Kuntīdevī, along with her children, was also saved from a series of dangers. But Kuntīdevī was shown far more favor because Lord Kṛṣṇa did not save the other children of Devakī, whereas He saved the children of Kuntīdevī. This was done because Devakī's husband, Vasudeva, was living, whereas Kuntīdevī was a widow, and there was none to help her except Kṛṣṇa.
(Srimad Bhagavatam----1:8:23----purport).

It is also said that Vasudeva and Devakī, in their previous birth as Sutapā and Pṛśni, underwent a severe type of penance to get the Lord as their son, and as a result of such austerities the Lord appeared as their son.
(Srimad Bhagavatam----1:8:33----purport).

Lord Kṛṣṇa appeared, just like electricity, when there was friction between Kaṁsa and Vasudeva and Ugrasena. Vasudeva and Ugrasena were the Lord's devotees, and Kaṁsa, a representative of the karmīs and jñānīs, was a nondevotee. Kṛṣṇa, as He is, is compared to the sun. He first appeared from the ocean of the womb of Devakī, and gradually He satisfied the inhabitants of the places surrounding Mathurā, just as the sun enlivens the lotus flower in the morning.
(Srimad Bhagavatam----3:2:15----purport).

After marrying Devakī, Vasudeva was returning home with her on a chariot driven by Kaṁsa, her brother, when an ominous voice addressed Kaṁsa, warning him thatDevakī's eighth son would kill him. Upon hearing this omen, Kaṁsa was immediately ready to kill Devakī, but Vasudeva diplomatically began to instruct him. Vasudeva stressed that it would not be good for Kaṁsa to kill his younger sister, especially at the time of her marriage. Anyone who possesses a material body must die, Vasudeva advised him. Every living entity lives in a body for some time and then transmigrates to another body, but one is unfortunately misled into accepting the body as the soul. If a person under this mistaken conception wants to kill another body, he is condemned as hellish.
(Srimad Bhagavatam----10:1--summary).

Because Kaṁsa was not satisfied by Vasudeva's instructions, Vasudeva devised a plan. He offered to bring Kaṁsa all of Devakī's children so that Kaṁsa could kill them. Why then should Kaṁsa kill Devakī now? Kaṁsa was satisfied by this proposal. In due course of time, when Devakī gave birth to a child, Vasudeva brought the newborn baby to Kaṁsa, who, upon seeing Vasudeva's magnanimity, was struck with wonder. When Vasudeva gave Kaṁsa the child, Kaṁsa, showing some intelligence, said that since he was to be killed by the eighth child, why should he kill the first? Although Vasudeva did not trust him, Kaṁsa requested Vasudeva to take the child back. Later, however, after Nārada approached Kaṁsa and disclosed to him that the demigods were appearing in the Yadu and Vṛṣṇi dynasties and conspiring to kill him, Kaṁsa decided to kill all the children born in these families, and he also decided that any child born from the womb of Devakī must be killed. Thus he arrested and imprisoned both Devakī and Vasudeva and killed six of their sons, one after another.
(Srimad Bhagavatam----10:1--summary).

The omen spoke of aṣṭamo garbhaḥ, referring to the eighth pregnancy, but did not clearly say whether the child was to be a son or a daughter. Even if Kaṁsa were to see that the eighth child of Devakī was a daughter, he should have no doubt that the eighth child was to kill him. According to the Viśva-kośa dictionary, the word garbha means "embryo" and also arbhaka, or "child." Kaṁsa was affectionate toward his sister, and therefore he had become the chariot driver to carry her and his brother-in-law to their home. The demigods, however, did not want Kaṁsa to be affectionate toward Devakī, and therefore, from an unseen position, they encouraged Kaṁsa to offend her. Moreover, the six sons of Marīci had been cursed to take birth from the womb of Devakī, and upon being killed by Kaṁsa they would be delivered. When Devakī understood that Kaṁsa would be killed by the Supreme Personality of Godhead, who would appear from her womb, she felt great joy.
(Srimad Bhagavatam----10:1:34----purport).

According to Vedic principles, a brāhmaṇa, an old man, a woman, a child or a cow cannot be killed under any circumstances. Vasudeva stressed that Devakī was not only a woman but a member of Kaṁsa s family. Because she was now married to Vasudeva, she was para-strī, another man's wife, and if such a woman were killed, not only would Kaṁsa be implicated in sinful activities, but his reputation as king of the Bhoja dynasty would be damaged. Thus Vasudeva tried in many ways to convince Kaṁsa in order to stop him from killing Devakī.
(Srimad Bhagavatam----10:1:37----purport).

Vasudeva wanted to save the life of Devakī by promising to deliver his sons to Kaṁsa. "In the future," he thought, "Kaṁsa may die, or I may not beget any sons. Even if a son is born and I deliver him to Kaṁsa, Kaṁsa may die at his hands, for by providence anything could happen. It is very difficult to understand how things are managed by providence." Thus Vasudeva decided that he would promise to deliver his sons to the hands of Kaṁsa in order to save Devakī from the imminent danger of death.
(Srimad Bhagavatam----10:1:49-50----purport).

Since the eighth son of Devakī was to kill Kaṁsa, one might ask what the need was for Vasudeva to deliver the first-born child. The answer is that Vasudeva had promised Kaṁsa that he would deliver all the children born of Devakī. Kaṁsa, being an asura, did not believe that the eighth child would kill him; he took it for granted that he might be killed by any of the children of Devakī. Vasudeva, therefore, to save Devakī, promised to give Kaṁsa every child, whether male or female. From another point of view, Vasudeva and Devakīwere very pleased when they understood that the Supreme Personality of Godhead, Kṛṣṇa, would come as their eighth son. Vasudeva, a pure devotee of the Lord, was eager to see Kṛṣṇa appear as his child from the eighth pregnancy of Devakī.
(Srimad Bhagavatam----10:1:58----purport).

After Kaṁsa killed the ṣaḍ-garbhas, the six sons of Devakī, one after another, Anantadeva entered Devakī's womb and was transferred to the womb of Rohiṇī by the manipulation of Yogamāyā, who was following the order of the Supreme Personality of Godhead. The Lord Himself, who was soon to appear as the eighth son of Devakī, ordered Yogamāyā to take birth from the womb of Yaśodādevī. Because Kṛṣṇa and His potency, Yogamāyā, appeared simultaneously as brother and sister, the world is full of Vaiṣṇavas and śāktas, and there is certainly some rivalry between them. Vaiṣṇavas worship the Supreme Lord, whereas śāktas, according to their desires, worship Yogamāyā in forms like Durgā, Bhadrakālī and Caṇḍikā. Following the orders of the Supreme Personality of Godhead, Yogamāyā transferred Baladeva, Saṅkarṣaṇa, the seventh child of Devakī, from the womb of Devakī to the womb of Rohiṇī. Because Saṅkarṣaṇa appears in order to increase love of Kṛṣṇa, He is known as Baladeva. One may take auspicious strength from Him to become a devotee of the Lord, and therefore He is also known as Balabhadra.
(Srimad Bhagavatam----10:2--summary).

After Yogamāyā transferred the seventh child of Devakī to the womb of Rohiṇī, the Supreme Personality of Godhead appeared within the heart of Vasudeva and transferred Himself into the heart of Devakī. Because the Lord was present in her heart, Devakī, as her pregnancy continued, appeared effulgent. Upon seeing this effulgence, Kaṁsa was full of anxiety, but he could not harm Devakī because of their family relationship. Thus he began indirectly thinking of Kṛṣṇa and became fully Kṛṣṇa conscious.
Meanwhile, because of the Lord's presence within the womb of Devakī, all the demigods came to offer the Lord their prayers.
(Srimad Bhagavatam----10:2--summary).

The first plenary expansion of Kṛṣṇa is Baladeva, also known as Śeṣa. The Śeṣa incarnation of the Supreme Personality of Godhead supports the entire universe, and the eternal mother of this incarnation is mother Rohiṇī. "Because I am going into the womb of Devakī," the Lord told Yogamāyā, "the Śeṣa incarnation has already gone there and made suitable arrangements so that I may live there. Now He should enter the womb of Rohiṇī, His eternal mother."
In this connection, one may ask how the Supreme Personality of Godhead, who is always situated transcendentally, could enter the womb of Devakī, which had previously been entered by the six asuras, the ṣaḍ-garbhas. Does this mean that the Ṣaḍ-garbhāsuras were equal to the transcendental body of the Supreme Personality of Godhead? The following answer is given by Śrīla Viśvanātha Cakravartī Ṭhākura.
(Srimad Bhagavatam----10:2:8----purport).

Symbolically, mother Devakī's constant fear of Kaṁsa was purifying her. A pure devotee should always fear material association, and in this way all the asuras of material association will be killed, as the ṣaḍ-garbhāsuras were killed by Kaṁsa. It is said that from the mind, Marīci appears. In other words, Marīci is an incarnation of the mind. Marīci has six sons: Kāma, Krodha, Lobha, Moha, Mada and Mātsarya (lust, anger, greed, illusion, madness and envy). The Supreme Personality of Godhead appears in pure devotional service. This is confirmed in the Vedas: bhaktir evainaṁ darśayati. Only bhakti can bring one in contact with the Supreme Personality of Godhead. The Supreme Personality of Godhead appeared from the womb of Devakī, and therefore Devakī symbolically represents bhakti, and Kaṁsa symbolically represents material fear. When a pure devotee always fears material association, his real position of bhakti is manifested, and he naturally becomes uninterested in material enjoyment. When the six sons of Marīci are killed by such fear and one is freed from material contamination, within the womb of bhakti the Supreme Personality of Godhead appears. Thus the seventh pregnancy of Devakī signifies the appearance of the Supreme Personality of Godhead.
(Srimad Bhagavatam----10:2:8----purport).

The word aṁśa-bhāgena is important in this verse. In Bhagavad-gītā (10.42) the Lord says:
athavā bahunaitena
kiṁ jñātena tavārjuna
viṣṭabhyāham idaṁ kṛtsnam
ekāṁśena sthito jagat
"But what need is there, Arjuna, for all this detailed knowledge? With a single fragment of Myself I pervade and support this entire universe." Everything is situated as a part of the Supreme Lord's potency. In regard to Lord Kṛṣṇa's appearance in the womb of Devakī, Brahmā played a part also because on the bank of the milk ocean he requested the Supreme Personality of Godhead to appear. A part was also played by Baladeva, the first expansion of Godhead. Similarly, Yogamāyā, who appeared as the daughter of mother Yaśodā, also played a part. Thus jīva-tattva, viṣṇu-tattva and śakti-tattva are all integrated with the Supreme Personality of Godhead, and when Kṛṣṇa appears, He appears with all His integrated parts. As explained in previous verses, Yogamāyā was requested to attract Saṅkarṣaṇa, Baladeva, from the womb of Devakī to the womb of Rohiṇī, and this was a very heavy task for her.
(Srimad Bhagavatam----10:2:9----purport).