It's a different sort of race with a lot of different things going on - you have the GC (yellow) of course, but there are teams who focus solely on getting the green or polka dot jerseys too. Young rider is an honor and of course, the previous day's most aggressive rider gets a fancy number.
In general, they want to make their team look good and winning a jersey and winning stages are the main ways but not the only way. Teams also form impromptu on-the-road alliances and work together for a myriad of goals, not all of them obvious.
One reason to get a rider in a breakway is so that your team leader has a helper way up front if he manages to bust off the front. A team mate is obviously a rolling trove of spare parts for the leader, so it's really critical that your leader has team mates as long as possible.
This whole business of following somebody up a mountain cracks me up - at the speeds I ride, there's no benefit, but those guys go fast enough to draft UP a mountain stage. Incroyable.