I'll expand the example sentence you gave to make the explanation easier:
"She graduated from New York University a few years ago. She went on to become a successful business woman."
The phrase "went on to" implies here that after she graduated from college, she later became a successful business woman. In general, "after she did X, she went on to do Y" it is basically just a way of saying "after she did X, she did Y".
However, this construct is not only used in reference to people. For example:
The book began by with an overview of basic English grammar. It then went on to discuss English internet slang.
Here, it is used to refer to the arrangement of a book's contents, rather than the actions/experiences of a person. But the pattern is basically the same. It's essentially just saying "first the book discussed X, then it discussed Y".