According to the BBC, the pulse rate increases after exercise because exercise increases the need for energy in the body, which in turn increases the need for food and oxygen. The breathing rate also speeds up during and after exercise to help transport oxygen and get rid of carbon dioxide.
The BBC points out that an increased pulse rate indicates the rate at which the heart is pumping and the arteries are expanding. During the recovery time after exercise, the pulse and breathing rates return to normal. A fit person or athlete experiences a smaller increase in pulse during exercise and a shorter recovery time afterwards.