I have chest pain, but only at the right side. is exercising at the gym, and supposedly a muscle issue a good explanation ?

Yes: Yes, at 21 years of age and exercising at the gym, it is very likely this is simply a muscle pull. Things that would make this more likely include if it is tender when you push on it and when you stretch the muscle. I would be more worried if these things are not true, or if you have other risk factors - such as a strong family history of early heart disease, smoke, etc...