Why do people stay in trans? Doesn't trans stand for transitioning gender? Wouldn't that mean that you're transition from one gender and not just staying in the body (gender) you were born in.

Yes it does. But there is a difference between Transsexual and Transgender. As you said Transgender is where you switch your original gender to a new one and Transsexual is when you wanna dress or look like the other gender but you don't want to switch it.