Even without the bishop on c5, White plays c3 all the time.
Historically speaking, the elixir for e4 players was to try to establish the classical pawn duo on e4 and d4. Therefore, they developed a lot of theory in the king's pawn opening, that all revolve around fighting for this pawn duo.
The approach is, of course, to play c3 so that when you play d4, you're prepared to recapture the pawn with the c3 pawn, instead of with the f3 knight or the queen.
Historically speaking, it has been decided that the Ruy Lopez is the best way to fight for this d4 and e4 pawn duo, because the Ponziani (1. e4 e5 2. Nf3 Nc6 3. c3) runs into some problems after 3...d5, neutralizing the center right away (which Bb5 doesn't allow) (Black has to know some theory, though. There are actually a lot of traps in the Ponziani!).
Now why is this e4 and d4 pawn duo better than say... Nc3? Well, the knight on c3 is actually quite passive. The knight belongs on g3, which is why the Ruy Lopez consists of a very common maneuver, that is, Re1 to prepare Nd2-Nf1-Ng3 (or Ne3!). Furthermore, having a pawn duo for white is always considered to make a difference between having an advantage and being equal.
Lastly, because the bishop is on c5 which it usually doesn't go to (goes on e7 or sometimes g7), the c3-d4 pawn break becomes even more attractive, hitting the bishop with tempo. However, some of the Bc5 lines are quite sharp and so White has to study some theory (it's called the Archangel!), and not always does white play c3-d4, but rather d3 (solidifying e4), followed by Nc3, Bg5, and Nd5. This is usually only when Black plays Bb7, though (and by playing d3, that would shut off a bishop on b7).
Edit: I forgot to mention that White also plays c3 so that the bishop can retreat to c2.