The problem is that it is one of the easiest things in the world to evoke an emotional response. A few notes on the piano will easily move one emotionally. A single look from another person can cause one to feel fear, lust, anger, peace, etc. A few lines of a poem can bring one to tears. A scene in a movie can evoke these same emotions.
The missionaries teach investigators that good feelings about some Church teachings witness that the teachings are true (source). But if one hears another doctrine of the Church that evokes a different emotional response, like disgust, that is apparently not a message from God, because it might lead one to conclude that the Church is not what it claims to be (more). Then, when someone such as I tries to investigate the real history and teachings of the Church, and finds strong physical, archaeological, or logical evidence that it is not what it claims to be, the Church insists that all those clues are irrelevant compared to the emotional conviction they have. Doctrine & Covenants, for example, states:
Verily, verily, I say unto you, if you desire a further witness, cast your mind upon the night that you cried unto me in your heart, that you might know concerning the truth of these things (Section 6:22).
In other words, if something seems like it's not right - if it seems like you need more from the Church to keep following it - what you are supposed to do is recall the one time you did feel good about it. If you feel bad about it now, there's nothing wrong with the Church, you just need to start feeling good again.
In brief, if you ever have a positive feeling about any part of the Church, it must be true. Any bad feelings you have about it are distractions from Satan. I've spent a very long time trying to come up with an adequate analogy, but it's so nonsensical that nothing fits. So here's a try:
Emotions are easily swayed, much like a feather moving in the breeze. So imagine a feather tied to a string, hanging from a tree branch. A man sits under the tree and watches the feather one day. A bird up on the branch tells the man that it is a magic feather - it can predict the weather. If the feather moves to the north, there is a thunderstorm coming. If it moves south, there will be sunshine. If it moves east, there will be snow, and if it moves west, there will be rain. The man watches for a while and the feather dances mostly Northwest. He sees some storm clouds off in the distance and concludes that the feather is magic. The next day, the feather sways eastward, but it stays sunny and warm. The man says, "Well, I already know it's a magic feather." So when he learns later in the day that it snowed in Alaska that day, his conclusion is confirmed. "The feather's magic is so powerful, that it could see it was snowing hundreds of miles away!" The next day, the feather moves to the south, and the sun shines. It is indeed a magic feather! The next day it blows to the west, but the sun shines again. A week later, it finally does rain and the man is awestruck that it not only predicted the weather, but that it did it a week in advance!
An easily-influenced variable is given ultimate authority. But when the easily-influenced variable acts unpredictably it is meaningless, or still evidence of the purported source's authenticity. This logic contains no real connection between what happens and what is real.