Imagine that society is a ladder. The people at the bottom of the ladder are poor. The people at the top are rich, powerful, and famous.
So when a person works hard to get promotions at work, make more money, and impress people, you can say that they are "climbing the social ladder".
This phrase has negative associations. We think of someone who "climbs the social ladder" as being a little bit greedy, sneaky, and aggressive.
Another common phrase with a similar meaning is "climb the corporate ladder", which means to get higher and higher positions in a large company.