bxc3 gives you a majority of pawns in the center, and support for the pawn break
d4!, possibly pushing
c4 and still having d4 pawn-supported. Big advantages in the long-term.
dxc3, you're right, the bishop is free, but with fewer pieces on the board, and Black to play, it's not much of an advantage. The file for the Queen, “attacking” the
♟d7, which belongs on
d6 where it is well defended, is worth nothing, and the possibility of castling queenside is not very relevant either (e.g. Black traded his f2- and h2-threatening bishop)
Also keep in mind that the c pawns are doubled for the time being, and you don't want this situation to last until the endgame. It will be a lot easier to trade the
♙c3 on the d file than trying to support the b pawn, or merely blocking the d file for Black.