Body Dysmorphia can be a part of being transgender, but it's categorised by an intense dislike of certain aspects of one's body and a desire to change them. The dysphoria people mention is Gender Dysphoria; it's an intense upset/dislike at being perceived as you assigned gender, when you identify as another gender to the one assigned. This is why changing your outward appearance can help.

If the issue were say your body, changing your clothes wouldn't make a difference, as you're body would be the same, but if the issue is how people treat you and talk about you and so forth, then changing your outward appearance can help people to see you as how you identify.

We live in a very binary society, so if you're upset by people seeing you as x, then it can be helpful to focus very much on presenting as y, and - hopefully - people will then treat you as y, which then alleviates the gender dysphoria. This isn't what makes someone transgender, though.

A transgender person is just someone that identifies differently to their assigned gender; some people can't pass, others aren't in a safe space to come out, others can't afford treatments, some are happy in their bodies and don't want treatments . . . all are very valid, but the core similarity is just that they share a conflict between assigned gender and actual identified gender.