I remember a couple of analogies that I read and heard as I was growing up. One was: "The Gospel is like a big jigsaw puzzle. Sometimes when you're working a puzzle, you can't see how to make all the sky pieces fit right away. That doesn't mean that you throw up your arms and say, 'This puzzle isn't true!' Sometimes you have to fit in other pieces first before you can see how it all fits together, but if you just have enough faith and keep trying, eventually, it will all fit." The other was: "Well, I don't have the slightest idea how a computer works, but that doesn't change the fact that it does work."
The message was, "Don't fret about the stuff that doesn't make sense. We can't possibly understand everything about God. As long you feel the Spirit, you know it's all true."
In his recent Ted talk, physicist David Deutsch defines a bad explanation: it is one that is too easy to vary. As soon as we have another explanation that could account for the experience of "feeling the Spirit" (and I believe that there are other explanations and intend to explore this topic in the future), the entire justification for believing in religion breaks down.
I've noticed something interesting since I quit believing in God: I still "feel the Spirit," or at least, I still have the feeling that as a Mormon I called "feeling the Spirit." I feel it when I'm holding one of my children, when I'm out in nature, sometimes when I'm doing yoga, and frequently when I'm creating or listening to music.
I do believe that it's a real experience. I don't believe that religion has a monopoly on it.