What is the purpose of checking your pulse rate after an exercise?

Correct zone. You need to exercise with enough vigor to raise your heart rate high enough for effective workout for heart and muscles - aerobic level. However, too high of a heart rate -"red zone" - can cause health problems especially if you have underlying problems. Most gyms have a poster of the ideal heart rate range based on your age.
Monitor conditioning. Many reasons. First of all there are different pulse zones for different goals. For example you can do better with weight loss at about 65-70% of max heart rate. Also good to follow conditioning progress. The quicker you get to your baseline pulse after exercise the more fit you have become.