Why do people get sore after working out?

Muscle soreness occurs because muscle and the connective tissue around it get damaged during exercise, such as weight lifting or high intensity training. This kind of natural breakdown is natural and needed for muscle growth, since muscle grows back stronger during the body's repair process. During this process the affected muscle fibers increase in strength and thickness. 

Additionally, people tend to experience lactic acid buildup prior to exercise as your body uses oxygen to convert glucose to energy. During exercise, if there is a lack of oxygen for this process, you may produce a substance called lactate. Many people believe that soreness comes from the buildup of this lactic acid, but the real contributor is the breakdown of muscle from usage.