Why does Black want to trade off her King's Bishop in many d4 lines? I feel that in many lines starting with d4, such as the Queen's Gambit Declined, or especially the Nimzo-Indian, Black is more than happy to trade her King's Bishop for White's knight on c3. For instance, one frequent line in the Nimzo-Indian (reached in 3200 games of the chess365 database) would be the following:



[FEN ""]
1. d4 Nf6 2. c4 e6 3. Nc3 Bb4 4. a3 Bxc3+ 5. bxc3



What strategic benefit does this hold for Black? She has given up the bishop pair, given White the chance to capture toward the centre with her b-pawn, in turn strengthening the d4 square for White, and moved the same piece twice in the opening. And yet, this seems to be a line where Black enjoys a higher win percentage than White, which puzzles me as it contradicts my naive understanding of opening principles.
Black gains a tempo. Black has played two bishop moves, but white has played Nc3, a3, and bxc3. Once the smoke clears, black has a lead in development with one minor piece out (vs none) and can immediately castle. Also, it doesn’t hurt that the doubled pawns are a weakness that can be exploited later. 

If white does not play a3, black has no immediate need to take the knight, and it makes more sense to keep the pin. This is especially true because black does not gain the tempo without a3.