Even if you are not transgender, just imagine changing your name, telling everyone about it, and someone would come along insisting on calling you using your former name, practicly mocking, belittling you and your choice. Now imagine them doing the same to your entire identity.
Or a daughter who dislikes her father so much that, upon reaching adulthood, changes her surname because she wants to be completely disassociated from him. (In the case I'm thinking of, she said something along the lines "he wasn't my father, he was the sperm donor" - he left when she was a few years old.)
Personally, I understood the idea first through Malcolm X, who rejected "the white slavemaster name of 'Little'" he was given at birth. And the respect Howard Cosell had to refer to Muhammad Ali by Ali's chosen name, while most others used and even taunted him with his old one.