Silence means that one is always thinking of self-realization. The person in Kṛṣṇa consciousness observes perfect silence in this sense.
(Bhagavad-Gita-------17:16-------purport).

One must observe silence, acquire steadiness by practicing different yogic postures, control the breathing of the vital air, withdraw the senses from sense objects and thus concentrate the mind on the heart.
(Srimad Bhagavatam------3:28:5-------translation).

Silence means talking only of kṛṣṇa-kathā. This is the silence of Mahārāja Ambarīṣa:
sa vai manaḥ kṛṣṇa-padāravindayor
vacāṁsi vaikuṇṭha-guṇānuvarṇane
"King Ambarīṣa always fixed his mind on the lotus feet of the Lord and talked of Him only." (S.B.9:4:19) We should also take this opportunity in life to become as good as a great saint simply by not talking unnecessarily with unwanted persons. We should either talk of Kṛṣṇa or chant Hare Kṛṣṇa undeviatingly.
(Srimad Bhagavatam------3:24:71-------translation).

Talkativeness can be avoided by gravity and silence, and by prowess one can avoid fearfulness.
(Srimad Bhagavatam------1:9:27-------purport).

Mauna means "silence." Unless one becomes silent, he cannot think completely about the pastimes and activities of the Lord. It is not that because one is a fool and cannot speak nicely he therefore takes the vow of mauna. Rather, one becomes silent so that people will not disturb him. It is said by Cāṇakya Paṇḍita that a rascal appears very intelligent as long as he does not speak. But speaking is the test.
(Srimad Bhagavatam------3:24:42-------purport).

Silence does not mean that one should not speak at all, but that he should not speak of nonsense.
(Srimad Bhagavatam------3:27:7-------purport).

The tongue is meant for speaking, although sometimes, to make a big show, a person remains silent. There are many who observe silence some day in a week. Vaiṣṇavas, however, do not observe such silence.
(Srimad Bhagavatam------7:9:46-------purport).

One who wants to say something can show himself to be a big orator, but rather than go on speaking nonsense, better to remain silent. This method of silence, therefore, is recommended for persons very attached to speaking nonsense.
(Srimad Bhagavatam------7:9:46-------purport).

In the Varāha Purāṇa the following offenses are mentioned: (a) to eat in the house of a rich man, (b) to enter the Deity's room in the dark, (c) to worship the Deity without following the regulative principles, (d) to enter the temple without vibrating any sound, (e) to collect food that has been seen by a dog, (f) to break silence while offering worship to the Deity, (g) to go to the toilet during the time of worshiping the Deity, (h) to offer incense without offering flowers, (i) to worship the Deity with forbidden flowers, (j) to begin worship without having washed one's teeth, (k) to begin worship after sex, (l) to touch a lamp, dead body or a woman during her menstrual period, or to put on red or bluish clothing, unwashed clothing, the clothing of others or soiled clothing. Other offenses are to worship the Deity after seeing a dead body, to pass air before the Deity, to show anger before the Deity, and to worship the Deity just after returning from a crematorium. After eating, one should not worship the Deity until one has digested his food, nor should one touch the Deity or engage in any Deity worship after eating safflower oil or hing. These are also offenses.
(Srimad Bhagavatam------7:5:23-24-------purport).

In support of their view that the self-realized remain silent, the Māyāvādīs are fond of using the example of the water pot, maintaining that when a pot is not filled with water it makes a sound, but that when it is filled it makes no sound. But are we waterpots? How can we be compared to them? A good analogy utilizes as many similarities between two objects as possible. A waterpot is not an active living force, but we are. Ever-silent meditation may be adequate for a waterpot, but not for us. Indeed, when a devotee realizes how much he has to say about Kṛṣṇa, twenty-four hours in a day are not sufficient. It is the fool who is celebrated as long as he does not speak, for when he breaks his silence his lack of knowledge is exposed. The Caitanya-caritāmṛta shows that there are many wonderful things to discover by glorifying the Supreme.
(Sri Caitanya Caritamrta------introduction).

When Sārvabhauma met Śrī Caitanya Mahāprabhu, he asked Him to hear Vedānta philosophy from him. Śrī Caitanya Mahāprabhu accepted this proposal, and for seven days He continally heard Sārvabhauma Bhaṭṭācārya explain the Vedānta-sūtra. However, the Lord remained very silent. Because of His silence, the Bhaṭṭācārya asked Him whether He was understanding the Vedānta philosophy, and the Lord replied, "Sir, I can understand Vedānta philosophy very clearly, but I cannot understand your explanations." There was then a discussion between the Bhaṭṭācārya and Śrī Caitanya Mahāprabhu concerning the authority of the Vedic scriptures, specifically the Upaniṣads and Vedānta-sūtra. The Bhaṭṭācārya was an impersonalist, but Śrī Caitanya Mahāprabhu proved that the Absolute Truth is the Supreme Personality of Godhead. He proved that the conceptions of the Māyāvādī philosophers concerning the impersonal Absolute Truth are incorrect.
(Sri Caitanya Caritamrta--------2:6------summary).

Lord Rāmacandra remained silent, and His silence shows that He accepted the prayers of the sages. Thus they were blessed by Lord Rāmacandra to have association with Lord Kṛṣṇa in their future lives. As a result of this benediction, they all took birth as women in the wombs of gopīs at Gokula, and as they had desired in their previous lives, they enjoyed the company of Lord Kṛṣṇa, who was present at that time in Gokula Vṛndāvana. The perfection of their human form of life was thus achieved by their generating a transcendental sentiment to share conjugal love with Lord Kṛṣṇa.
(Nectar of Devotion).

First one must control his speaking power. Every one of us has the power of speech; as soon as we get an opportunity we begin to speak. If we do not speak about Kṛṣṇa consciousness, we speak about all sorts of nonsense. A toad in a field speaks by croaking, and similarly everyone who has a tongue wants to speak, even if all he has to say is nonsense. The croaking of the toad, however, simply invites the snake: "Please come here and eat me." Nevertheless, although it is inviting death, the toad goes on croaking. The talking of materialistic men and impersonalist Māyāvādī philosophers may be compared to the croaking of frogs. They are always speaking nonsense and thus inviting death to catch them. Controlling speech, however, does not mean self-imposed silence (the external process of mauna), as Māyāvādī philosophers think. Silence may appear helpful for some time, but ultimately it proves a failure. The meaning of controlled speech conveyed by Śrīla Rūpa Gosvāmī advocates the positive process of kṛṣṇa-kathā, engaging the speaking process in glorifying the Supreme Lord Śrī Kṛṣṇa. The tongue can thus glorify the name, form, qualities and pastimes of the Lord. The preacher of kṛṣṇa-kathā is always beyond the clutches of death. This is the significance of controlling the urge to speak.

(Nectar of Instruction).