Ok, hear me out: the issue can be divided up into three questions: why did Tauriel like Kili? why did Kili like her back? is it even possible for an elf and a dwarf to be in love? So I'll start by addressing the last question first.
Are there any recorded instances of a romantic elven-dwaren relationship? No. But there is an instance in the Book of Lost Tales part II where the idea is vaguely suggested. So it's not entirely ludicrous.
Why did Tauriel like Kili? Perhaps she had never seen dwarves before, and finding that at least Kili didn't fit the dwarf stereotype among her people, took an interest in him that developed into a crush. Or, as an alternative reason, what girl doesn't have a crush on Aiden Turner, am I right?
As for why Kili liked Tauriel back, we see that in the extended edition of An Unexpected Journey, Kili has something of an attraction towards elves. But why, then? Maybe he's secretly half human? Or maybe another reason why he might have liked her is because she showed him kindness, and interest during his imprisonment. I certainly know that if I was in prison and a redhead girl who looked like Evangeline Lilly started giving me special attention, I'd probably get at least a very surreal feeling.
So, in my answering this, I had to use some fairly obscure logic. But I've made my case.