Okay kiddo, so a "white-shoe" firm is a special kind of company that has a long history of being very fancy and respectable. It's like the grown-up version of being part of a really fancy club.
These kinds of firms usually specialize in areas like law or finance, where it helps to be seen as really serious and important. They are called "white-shoe" firms because they used to require employees to wear shoes that were a certain color - you guessed it, white!
Today, being a part of a white-shoe firm means that you are part of a very exclusive club of highly-educated, well-connected, and successful people. These firms only hire the best of the best, which means that they can charge a lot of money for their services.
So basically, a white-shoe firm is like a super fancy club for super smart and successful people who work in important jobs like law and finance.