What is this solar power RC circuit doing?
I don't see the "R" in your RC circuit.

I see a capacitor and a varistor doing separate jobs, right next to one another.

C1 is there to provide current for a little while when there's a shadow on the solar cell.  

It also provides current when your load needs more current than the solar cell can provide - short bursts, of course.  It can't provide high current for long, but often times you just need a little burst to get something going - starting a motor or switching a relay, for example.

The varistor is there to protect your five volt load from over voltage from the solar cell.  

As the voltage rises above 5V, the varistor will start to conduct. Somewhere around 5.6V it will turn into a short circuit.  Depending on how much current the solar cell can deliver, it will either prevent the voltage from getting above 5.6V or it will blow the fuse.

Either outcome protects your 5V load from damage.

Without knowing what else is on the "USB"  line, there's no way to tell whether or not you could use a smaller capacitor for C1.  

There might be something there that needs a lot of current once in a while.  It might also be there just because one of the other folks in your group has a bad tendency to stand in the wrong spot and block the light from getting to the solar cell.



That is a strange way to do things.  It would be better to have a buck/boost switching regulator in there to maintain a proper 5V supply as long as the solar panel is providing enough power.