Genesis 6:2-4, Enoch 6-11, and logic. For the latter, one telling difference is the linguistic differences between biblical descriptions of Angels and Demons. Also, Angels (even Fallen angels) are able to manifest themselves with man-like bodies, yet Demons are always desperate for habitation. There's also the fact that, while even Michael would not rebuke Satan (clearly a fallen Angel) when fighting for Moses' body, we are encouraged to rebuke demons when we encounter them.
There's a lot of evidence, but it's "scary" so modern churches don't talk about it any more than they talk about sin or damnation. But this is a ruling thread, and I've hijacked enough for now 
In your believe, then, where did the demons come from (just curious)?
According to the book of Enoch (non-canonical, but still the only source we have that talks about this kind of thing), demons are the disembodied spirits of the Nephilim. This would make logical sense as well if you follow a literal interpretation of Genesis 6:2-4. Clearly the Nephilim are not fully human, so what would happen to their soul following death is a mystery, except when you factor in these "demons" which are otherwise unaccounted for, and it makes sense.
