Well, I'm not sure I have the whole story, but I just got a lot more of it yesterday...
Quick Spanish lessonLet's start with Spanish. In Spanish, there are two different words for "you": tú and usted. "Tú" is informal/familiar, and is what you'd use when talking to a child (or someone obviously younger than you), or when talking to a close friend. "Usted" is a more formal term, and is used to show respect, deference, and manages to maintain a reasonable "distance" between you and the other person.
When learning how to pray in Spanish, I figured we'd use the "usted" when speaking to our Heavenly Father, but turns out the "tú" form is what's used. Why? As it was explained to me, our communication/communion with our Father should reflect a very close, loving, familiar relationship, which is what the tú form reflects.
Et tu, English?So, back to English -- why "thou" and not "you"?
According to Grammar Girl, back in the 14th century English used to have formal and familiar forms of "you" too... But what surprised me is that when we went down to one, we dropped the familiar form! Yep, that's right, "thou" is the familiar/informal form, and "you" is the formal form.
So when you say "thee" and "thou", you're actually using the less-formal, familiar-relationship forms of you.
Learn something every day.
* Interesting sidenote, as GG points out, "you" is both a subject *and* object pronoun, e.g. "You are very kind.", and "I gave the books to you." But the familiar versions of you have different subjective and objective forms, e.g. "Thou art most kind." and "I trust thee."