I listened to the provided video, and I feel you miss the key point:
You will never know a spiritual truth except it is revealed to you by the Holy Spirit. This is what Elder Ballard wants to say.
He is not saying that every feeling we get comes from God. He is also simplifying a little bit to get his point across.
You question of "What does it mean that religious truth is confirmed by what you feel?" is hard to answer, because what that means varies greatly. What he actually wants to say is: Physical Evidence will never be enough to produce faith or to find spiritual truth. Lack of physical evidence is no excuse for lack of faith, because God does communicate with us. But how this communication expresses itself varies from person to person, from circumstance to circumstance.
Scriptural examples of how physical evidence is not enough to understand spiritual Truth:
26 Six days ye shall gather it; but on the seventh day, which is the
sabbath, in it there shall be none.
27 ¶And it came to pass, that there went out some of the people on
the seventh day for to gather, and they found none.
28 And the Lord said unto Moses, How long refuse ye to keep my
commandments and my laws?
29 See, for that the Lord hath given you the sabbath, therefore he
giveth you on the sixth day the bread of two days; abide ye every man
in his place, let no man go out of his place on the seventh day.
30 So the people rested on the seventh day.
Excuse me, are these the people that God just showed mighty miracles, freed from slavery, and pretty much showed himself to? Yet some don't trust what he says...
1 Nephi 17:45
45 Ye are swift to do iniquity but slow to remember the Lord your
God. Ye have seen an angel, and he spake unto you; yea, ye have heard
his voice from time to time; and he hath spoken unto you in a still
small voice, but ye were past feeling, that ye could not feel his
words; wherefore, he has spoken unto you like unto the voice of
thunder, which did cause the earth to shake as if it were to divide
Nephi's brothers had seen an angel, yet they continually doubted.
Related to this:
1 Nephi 15:7-11
7 And they said: Behold, we cannot understand the words which our
father hath spoken concerning the natural branches of the olive tree,
and also concerning the Gentiles.
8 And I said unto them: Have ye inquired of the Lord?
9 And they said unto me: We have not; for the Lord maketh no such
thing known unto us.
10 Behold, I said unto them: How is it that ye do not keep the
commandments of the Lord? How is it that ye will perish, because of
the hardness of your hearts?
11 Do ye not remember the things which the Lord hath said?—If ye will
not harden your hearts, and ask me in faith, believing that ye shall
receive, with diligence in keeping my commandments, surely these
things shall be made known unto you.
Heck, a good portion of the witnesses of the Book of Mormon, who saw the gold plates, fell away (some returning, some not, as far as I am aware). Without revoking their testimony concerning the plates, mind you.
As to how spiritual Truth is confirmed, there are a few scriptures I can cite, but I agree with Jake Toronto who commented that discerning what is spiritual communication and what not is essentially a life-long journey.
Doctrine and Covenants 8:2
2 Yea, behold, I will tell you in your mind and in your heart, by the
Holy Ghost, which shall come upon you and which shall dwell in your
Doctrine and Covenants 50:12-24
12 Now, when a man reasoneth he is understood of man, because he
reasoneth as a man; even so will I, the Lord, reason with you that you
13 Wherefore, I the Lord ask you this question—unto what were ye
14 To preach my gospel by the Spirit, even the Comforter which was
sent forth to teach the truth.
15 And then received ye spirits which ye could not understand, and
received them to be of God; and in this are ye justified?
16 Behold ye shall answer this question yourselves; nevertheless, I
will be merciful unto you; he that is weak among you hereafter shall
be made strong.
17 Verily I say unto you, he that is ordained of me and sent forth to
preach the word of truth by the Comforter, in the Spirit of truth,
doth he preach it by the Spirit of truth or some other way?
18 And if it be by some other way it is not of God.
19 And again, he that receiveth the word of truth, doth he receive it
by the Spirit of truth or some other way?
20 If it be some other way it is not of God.
21 Therefore, why is it that ye cannot understand and know, that he
that receiveth the word by the Spirit of truth receiveth it as it is
preached by the Spirit of truth?
22 Wherefore, he that preacheth and he that receiveth, understand one
another, and both are edified and rejoice together.
23 And that which doth not edify is not of God, and is darkness.
24 That which is of God is light; and he that receiveth light, and
continueth in God, receiveth more light; and that light groweth
brighter and brighter until the perfect day.
Let me tell you that sometimes this understanding means knowledge, sometimes it means insight (I lack a proper translation of the german word I mean, I hope this one hits the intended meaning). Insight, for me, means an understanding that can't necessarily be communicated. For example, I gained a little insight in the power of God when contemplating just how huge and powerful the sun is, which God created. That was a powerful experience, but none that I feel can be properly communicated, because the result was mostly feeling, and is directly coupled to the experience.
So, I hope this shows that spiritual communication is not "just" feelings. But it is also not "just" knowledge, in the intellectual sense. (And really, how do you expect Truth to be confirmed, if not also through feelings? He doesn't want intellectually "convinced" people, he wants spiritually "converted" people) In the context Elder Ballard said what he said, I feel it was appropriate to word it like he did.