Taking with a pawn, dxc6 or dxc3.
When I first began playing chess, I learned that one of the goals was to build a strong center, and the central pawns are therefore considered more valuable than the outer pawns. So, with this knowledge, it always confused me when I saw exchanges on c6 or c3 where the defending player would recapture with the d-pawn rather than the b-pawn. The reason, I have learned, is to open up attacking lines and get compensation for the doubled pawns and missing d-pawn through rapid development and dynamic attacking chances. Here is an example (though, granted, my opponent didn't play the most accurate moves, and I'm sure I didn't either).
Any thoughts? I am not a strong player, as my rating will attest. I'm here to learn and get better, so anyone with some tips or more about capturing with a central pawn rather than an outside pawn for development purposes would be a welcome addition.