The owner maintains the car, but Uber has to pay the driver enough to cover the cost. So it's a cost for Uber even though it's not on their balance sheet. "The rider pays the driver" is just semantics since Uber decides the percentage.
Well the idea (originally) was that the driver is already driving and so they are sharing/renting out their open seat to get a little bit of money. Uber would just skim off the top for matching riders with drivers.
The reality of the economics is likely different but that's how it works in theory. Now they are likely paying drivers to be out looking for rides to ensure that service isn't disrupted, and yes they need to subsidize that as others are mentioning.
That's the idea at least.