Black's main threat after 3... c6 is d5; it looks like Qe2 is an attempt to prevent that, by indirectly attacking pawn e5, even with check. However, I'm not so sure that it works as intended, because Black can play d5 anyway, e.g.
1. e4 e5 2. Bc4 Nf6 3. d3 c6 4. Qe2 d5! 5. exd5 cxd5 6. Bb5+ Nc6 7. Qxe5+ Be7 8. Nf3 O-O 9. Bxc6 bxc6 10. O-O Bd6
and Black's center control, slight development advantage and bishop pair provide more than enough compensation for the pawn.
I'd prefer a normal developing move (4. Nc3 or Nf3) instead of 4. Qe2.