Daniel Negreanu would say 'He's probably got J9, I call'. Haha. I don't know what level of knowledge you have, so (to benefit the most people) I'll proceed on a fundamental basis. Sorry if you know some of this stuff but better safe than sorry!
Pre-flop: Can't fault what you've done pre-flop. Good raise, good sizing. Let's think about the range BTN has here. We have no reads, so it'll have to be a generic range. 22-JJ, ATo-AQo, A2s-AJs, KJo+, K6s+, QJo, Q9s+, J8s+, 54s+, 64s+, T7s.
Flop: It's a decent flop for us. We can get called by draws and pocket pairs, so a bet of about $0.20-$0.22, as you've done, is perfect. When he calls, we can narrow his range down. I've got 47 combos that make sense. TT-JJ (12 combos), 8x (6 combos, I imagine he folds some of the weaker ones), straight draws and flush draws (20 combos, mainly flush draws), 6 combos of 9x (including 98). I've taken a lot of 9x combos out because I think he raises 9x some of the time on the flop, and the same with some flush draws. So we're ahead of the majority of his range.
Turn: I don't think the Th is a bad card. It doesn't improve many of his hands that weren't already winning, except TT and 87. QJ overtakes you as well but you block that very well. You can still get value from flush draws and JJ, and you're rarely behind. I'd bet small, about $0.38 to $0.40 and fold to a raise. As played, after the check, we call this bet. He could be betting a flush draw (to get us to fold a hand like AK), or a hand like T8 or JJ. He could also be betting TT or 9x, but as I said, I think he often raises 9x on the flop.
River: Great card for us. For the people who are saying to raise his bet, he's put us all-in, so we can't. This is a snap call though. He can make this bet with 9x, and sometimes be bluffing a missed flush draw because it is a scary board if he put us on AA, or similar. Of course, he has a boat fairly often too, but it's still a call.
If I had to guess what he has here? TT.