What is the meaning of the phrase “clean up after” and when and how to use it?

Question

The guarantees made by constructors and destructors offer a nice way to let the compiler clean up after you.

Answer 1

From the Macmillan dictionary:

clean up after someone: to clean a place after someone has made it dirty or messy

Residents have been told to clean up after their dogs.

Another example (closer to the one in your question), desctructor functions in some programming languages let you remove from memory the variables that are no longer needed by the computer program, i.e. to perform a clean up when the program no longer needs to use an object.