In the past, topology was little more than an amusing diversion for mathematicians doodling about the difference between donuts and dumplings.
...what?
...what?
(And perhaps some areas of solid state do?)
EDIT: fixed mistake.
You see electromagnetism, realize that electromagnetic field is just 2-form on 4-manifold, and see that Maxwell equations just state that both this form and its Hodge dual are closed, so that they represent de Rham cohomology classes. Of course, you can also represent this result in a classical way, but creating a bridge translating physical concepts into well studied mathematical frameworks has the advantage enabling you to also pass the bridge in the other direction: sometimes you can find physical interpretation concepts that arised in the abstract setting.
So, we know that forms representing electromagnetic fields are closed, but what does it mean in physics when they are exact? How to interpret Mayer-Vietoris sequence in electromagnetism terms? What the induced maps do with forms representing electromagnetic fields, and how to interpret homotopy? What about Poincare duality?
It's such a great feeling to realize that your favourite toy, after playing with it for years, is actually able to do a whole lot more stuff than you were previously aware of.
The point is this -- if you want to extend our model of the universe, knowing the more abstract machinery is useful; by generalising your understanding (and a hell of a lot of math is basically generalising to more abstract systems) you gain insight into how alternative models might be constructed. If you want to understand our current model, even at a quite deep level, you can get by without knowing any topology at all.