What is the basis for the LDS belief that religious truth is confirmed by what you feel? LDS Elder M. Russell Ballard in a Youtube video said that


  Religious truth is always confirmed by what you feel. And that's the way heavenly father answers prayers.


What does it mean that religious truth is confirmed by what you feel? Are there any proofs for this argument? If yes, what are they?
Elder Ballard is talking about the fruit of the Spirit - the &quot;fruit&quot; being a manifestation, consequence, or result of the presence of the Holy Ghost.
This teaching is founded on a few scriptures, mainly Galatians 5:22-23 which says:

22 But the fruit of the Spirit is love, joy, peace, longsuffering, gentleness, goodness, faith,
23 Meekness, temperance: against such there is no law.

and Ephesians 5:9 which says:

9 (For the fruit of the Spirit is in all goodness and righteousness and truth;)

Spiritual truths are confirmed by spiritual means. (I recommend perusing the answer at that link for an explanation.) Confirming feelings come by revelation from God and cannot be coerced or forced upon another to prove things to them.
However, when someone speaks what the Holy Ghost inspires (a testimony), the Spirit can manifest truth of it to those receiving it:
2 Nephi 33:1

... when a man speaketh by the power of the Holy Ghost the power of the Holy Ghost carrieth it unto the hearts of the children of men.

The Book of Mormon promises revelation to those who seek truth in several instances, for example:
2 Nephi 32:5

5 For behold, again I say unto you that if ye will enter in by the way, and receive the Holy Ghost, it will show unto you all things what ye should do.

In more modern times, Joseph Smith received a revelation about conducting church meetings:
D&amp;C 46:7

7 But ye are commanded in all things to ask of God, who giveth liberally; and that which the Spirit testifies unto you even so I would that ye should do in all holiness of heart

The ultimate promise of revelation in the Book of Mormon is found in its closing verses as Moroni prepares to bury the record. It is probably the most-quoted passage by Latter-day Saints about how to know the truthfulness of the Book of Mormon:
Moroni 10

3 Behold, I would exhort you that when ye shall read these things, if it be wisdom in God that ye should read them, that ye would remember how merciful the Lord hath been unto the children of men, from the creation of Adam even down until the time that ye shall receive these things, and ponder it in your hearts.
4 And when ye shall receive these things, I would exhort you that ye would ask God, the Eternal Father, in the name of Christ, if these things are not true; and if ye shall ask with a sincere heart, with real intent, having faith in Christ, he will manifest the truth of it unto you, by the power of the Holy Ghost.
5 And by the power of the Holy Ghost ye may know the truth of all things.