I moved Qc2 because if my opponent had taken his bishop to e4, he would have forked my knights. How can the suggested 6.a4 move be better than guarding the e4 square?

The problem with Qc2 is that you have weaknesses along the a8-h1 diagonal. Your knight on c3 is now completely unguarded. If your opponent had responded to Qc2 with Nf6 your e4 pawn is still in danger and because of your move Qc2 it is now pinned. So, Qe2 would have been a better choice. The move a4 chooses to counterattack rather than protect. Trading the e4 pawn for the  b5 pawn might be ok here because it weakens the c4 pawn.