To move up in the hierarchy of a corporation.

A 'ladder' is a device with steps used to 'climb' (or move) up and down, so the 'corporate ladder' is the series of steps people go through as they gain more power in a corporation and 'rise to the top' - from file clerk up to president.

Referring to the heavy unfashionable winter jacket and patterned mittens Sanders wore to the inauguration of Joe Biden.