why are tires not a solid piece of rubber, and are instead hollow, and filled with air?
There are two main reasons for why tires are hollow and filled with air rather than solid rubber: 1) air can be compressed and adjust its volume more easily than a material like rubber, and 2) when traveling at high speeds, striking a small obstacle is absorbed better by a tire filled with air compared to a solid rubber tire [1, 2]. For example, when travelling at 60 mph, striking a small obstacle is absorbed by a tire filled with air (complemented by the shock absorbers installed in modern cars), whereas a solid rubber tire would absorb and pass along the shock of striking the obstacle, making for an extremely bumpy and uncomfortable ride [2]. However, at high speeds, the amount of energy loss while passing over bumpy roads would be extremely inefficient for the car, and would cause tires to wear down much more quickly [3]. Overall, tires are hollow because air can be compressed and adjusted to absorb and adjust to deformations more easily than a solid material like rubber [2]. 
Tires are typically made up of air because it is able to be compressed, which allows it to dampen the shocks of traveling over uneven surfaces [1, 2, 3]. In comparison, tires made of solid rubber would absorb the shock from the bump and pass it along to the vehicle, resulting in an extremely bumpy ride [3].