Why are homeomorphisms important?

Homeomorphisms are important because they are instances of a more general idea: structure-preserving isomorphisms. You will learn to appreciate this idea as you study more advanced mathematics.

In many domains of mathematical inquiry, the objects of study carry important kinds of "structure," and we don't care to distinguish two objects so long as they have the same structure. We can make the notion of "having the same structure" precise by saying two objects X and Y have the same structure precisely when there is a bijection between them that "preserves the structure" (in a sense that can also be made precise).

Homeomorphisms are precisely those functions for topology. Their cousins are group isomorphisms in group theory, and ring isomorphisms in ring theory, bijective linear transformations in vector space theory, etc.

If they are homeomorphic, they have the same topological properties. Topologically, they are the same, thus the joke that a topologist cannot tell apart the doughnut from the coffee mug. They are the same. Intuitively, you can transform a surface into another without making any tearings in the surface. If your doughnut is a muffin, without a hole, you cannot transform it into the coffee mug (or the doughnut) without making a hole, thus breaking the structure. The muffin is not homeomorphic to the coffee mug, that's why you should never have muffins with your coffee.

The notion of homeomorphism is of fundamental importance in topology because it is the correct way to think of equality of topological spaces. That is, if two spaces are homeomorphic, then they are indistinguishable in the sense that they have exactly the same topological properties.