Teh BB checked preflop, wich doesn' give us any info on his hand (and can put him on a broad range of hands). With no flush or straights possible on the board, you are either way behind or way in front:
a) BB has JJ and you've been beaten
b) BB has a 3 and a lower kicker and you're way in front
c) BB has 5 and J and you're way in front
d) BB has 3 and J and has flopped a full
e) BB has pocket 5's and you're way behind
Now, I think that it's someone messing with top pair and middle pair, but it very well could be a full or another set of trips with lower kicker. But the re-re-raise action looks like someone getting caught with his hand in the cookie jar and trying to get away with it (Jack with high kicker or two pairs).
If he had caught the trips on the flop with a 3, I expect either a check/raise either a check/call: he cannot suspect other trips and is slowplaying. But it looks way too much like a feller bet and feeling weakness, he decides that with J-5, he has the better hand at the turn.
I call, since I stand a very good chance of doubling up unless he has flopped a full house, wich is even more unlikely because betting first with trips when you have not much draws on the board, would've gotten you out of the pot if you hadn't hit the flop. Hard to tell without tells
from previous hands. That 0.25 (just like the .50 on the turn) looks like a bet from someone with two high cards... On the other hand, if the guy is a rock, I fold without hesitation, a maniac or a donk, I call.
PS: raising on the flop would've been wise because you are coming into the turn practically blind.