That "feel good" chemical is known as Endorphins. They are released during exercise because of the body goes through extended amounts of stress when exercising. Have you ever heard of the "Runners high"? It's actually a real sensation that you can thank Endorphins for.
Serotonin is also an important chemical that contributes to why exercise feels this way. Have you ever felt depressed after sitting around and doing nothing? Low serotonin is one of the biggest contributors to feeling depressed. Serotonin is in alot of anti-depressant medications. Serotonin is mainly released through exercise.
There could be a reason not relating to chemicals at all. Working out and seeing the results gives us a major confidence boost.