The move 4. c3 does two things:
- it supports the white center
- it states that white is not willing to take on c5
It is especially the latter, which is crucial. If white doesn't want to take and keep his centre intact black can play a "sort-of french" setup but without blacks biggest shortcoming there: the light-quared bishop and its lack of options. If white is not willing to take on c5 on move 4 he won't do so on move 5 or 6 either, hence:
1. e4 c6 2. d4 d5 3. e5 c5 4. c3 Nc6 5. Nf3 Bg4 6. Be2 e6!?
Develop the bishop c8 (to f5 or, after Nf3, to g4), only then play e6, arriving at a "french defense, advance variation", but without the bad bishop. The rest of the pieces develop like in the french defense: Nb8 goes to c6, Ng8 goes to e7 and f5 eventually (in rare cases to g6), Bf8 to e7, etc.. The plans are similar too: put pressure on the white centre with the levers c5 (this is already played) and f7-f6. If the bishop g4 is attacked or questioned with h3 it is probably best to exchange it for the Nf3, thereby diminishing whites control of the dark squares in the center (d4, e5). Retreating it to g6 would take possible squares away from the Ng8 to go to. Also, expand on the queen side where you have a space advantage.