To "use" can mean (though in most contexts this meaning is obsolete) to do something regularly or habitually. So "I use to do X" means "I am in the habit of doing X", and "I used to do X" means "I was in the habit of doing X". The implication that you don't do it any more is a Gricean thing -- if you were still doing it you'd say "I use to do X" rather than "I used to do X".
Nowadays no one uses "use to" in the present tense and no one is thinking of the above when they use "used to" in the past tense. But that's where it comes from.
[EDITED to add:] I see that this has got some downvotes, but I'm having trouble figuring out why. If you read this and didn't like it, I'd be very grateful if you let me know what problem(s) you see. Thanks!
(Of course maybe it's just random drive-by haters, but more likely there's something it would be useful for me to know. Maybe something I wrote is wrong, in which case I want to know. Maybe to some readers it looks like I'm insulting the person I was replying to or something, which wasn't at all my intention and if that's going on then I should probably clarify or apologize or something. I dunno.)