Set theory is important because it is a theory of integers, models of axiom systems, infinite ordinals, and real numbers, all in one unified structure. This allows it to serve as a foundation for all of mathematics, anything you talk about in mathematics can be formalized in set theory naturally and easily, and studying set theory allows you to prove theorems about mathematics itself. The formulation of set theory in the late 19th century motivated the metamathematics of the 20th century, with all the astonishing results about provability.
It is an extremely important subject, and I would recommend to read Paul Cohen's book "Set Theory and the Continuum Hypothesis", together with some historical work from the late 19th century or early 20th century, like Frege and Cantor, to see where the ideas are coming from, and further work from more recent authors, like Saharon Shelah, who is a big name with big theorems and big books.
