It is a mathematical device to define what a continuous function is, in a general setting.
A topological space is a set X together with a collection of subsets of X, called open, such that finite intersections of open sets are open, and arbitrary unions of open sets are open.
A function of topological spaces is continuous if inverse images of open sets are open.
What is the intuition behind (1) and (2)?
I claim that it is better to ask, instead, how mathematicians came up with (1) and (2).
