Not sure if you're really specifically asking about react? This question is maybe just "why render client-side, vs sever-side"? In the end you have to think about where code is stored, transferred and executed.
Client side rendering pushes app code down to the client (so there's more code to download, execute, etc; but it can be faster for subsequent actions), while server-side executes app code and only sends the resulting view down the pipe (which is lighter, but you go back to the server for each view). Neither solution is perfect, hence the various ways people are trying to get the best of both worlds.
I can see server-side rendering of React getting more popular simply to have a templating option that can live in both worlds. Even if you are just doing server-side rendering for one of your systems while the rest are single-page apps, it's nice to be able to use the same template set.