The goal is usually to avoid PKs -- which are exciting, but reduce the game down to the equivalent of a home run derby and are a pretty crap way of determining which team is "better." There have been a lot of fun suggestions on how to break ties (my favorite was playing an extra time period with the goalkeepers off the pitch), but the away-goal rule has been the go-to for most 2-leg matches around the world.
The problem I have with it in MLS is that it kills the idea of homefield advantage a bit and encourages negative play -- since the lower-seeded team can bunker for the 0-0 draw and set up the scenario where the higher-seeded team loses on draws in the second leg (since a 1-1 or 2-2 result would have the lower-seeded team going through on away goals). The away goal rule kinda works in a CL setup, where there hasn't been a full season of games that should mean something in terms of advantages in the playoffs, but it's bad in MLS.