It means the person who was learning something has become very very good at that thing, so good that they are probably better than their teacher. You can use this phrase when you do something better than the person who was supposed to be better than you. For example, if you take chess lessons and you beat your tutor, you can then say, "The student has become the master."

That means that the student is now more skilled or knows more than the teacher.

As the Chinese say, 青出于蓝。