Why does radioactive decay occur?
The simplified version of why radiation occurs is a balance between attraction of the protons and neutrons in the nucleus and the positive charge of the protons repulsing each other. Get the balance right, and you end up with a stable isotope, get it wrong, and the isotope is unstable and will decay.

As for the exponential decay: a nucleus falling apart is stochastic process, meaning it is only dictated by chance! When you take an unstable nucleus, after it's half-life there is a 50% chance that it will have decayed. Then, you can add more nuclei the chance of half of them decaying after it's half life is far, far larger than all of them being stable. The larger amount you start of with, the better this theory describes the decay. If you then measure a vast amount of atoms (a mole for example) the curve you find will describe an exponential decay.