I'm one of the co-founders of ShareLaTeX, and there are two main reasons that we've found that people really like ShareLaTeX. The first has been mentioned - you don't need to set up a LaTeX repository yourself. Even more than that, you can work on LaTeX documents from anywhere. It tends to be hard to find LaTeX installed on a computer if you're on the move and not using your own machine.
The other advantage is the ease of collaboration. While there are plenty of people in the LaTeX world who know git, there are plenty that don't as well. ShareLaTeX gives everyone a point and click way to make sure they are in sync with their collaborators. That said, we're getting quite close to offering git access to the projects for people who want it.
Also, my apologies that we keep appearing on the front page (along with a few other similar sites). We're flattered that people keep submitting us though.
I think the most important thing is you don't have to setup the compiler yourself. It may get out of hand pretty quickly if your partners and you don't have compatible environments (Windows vs. Linux, etc.).
