Topological spaces are good abstract spaces to study limit and continuity, just as vector spaces are good abstract spaces to study linear combinations. Like many abstractions, proofs are studied in less abstract settings (e.g., $\mathbb{R}$) to see what makes them tick.
So why open sets?
Topology is defined in terms of open sets because that formulation was introduced (by Hausdorff?) at just the right time to become popular and drive out any competing formulations. There are quite a few equivalent formulations: closed sets; neighborhoods; operation of taking interiors; closure operation; predicate that says when a point is a limit point of a set; and so forth.