I once saw a statement from Rav Hutner zt”l on this. He was a slick Slabodka dresser and dressed chassidish after the war. He said that
1. It actually started with Rav Yisroel Salanter, who felt that the bochurim were looked down upon by baalei battim and needed to look like they weren’t nebs
2. He changed because he wanted to show the world what a Jew looked like after churban Europe which wiped out most of the Jews.
Also, like Berel Wein said, the Mussar Movement lost. So white shirts black pants it is.
To be clear, most people aren’t wearing that to make a statement. They wear what everyone else wears. The uniform of a Ben Torah today is white shirts black pants.