Why is compactness so important?

As many have said, compactness is sort of a topological generalization of finiteness. And this is true in a deep sense, because topology deals with open sets, and this means that we often "care about how something behaves on an open set", and for compact spaces this means that there are only finitely many possible behaviors.

But why finiteness is important? Well, finiteness allows us to construct things "by hand" and constructive results are a lot deeper, and to some extent useful to us. Moreover finite objects are well-behaved ones, so while compactness is not exactly finiteness, it does preserve a lot of this behavior (because it behaves "like a finite set" for important topological properties) and this means that we can actually work with compact spaces.

The point we often miss is that given an arbitrary topological space on an infinite set $X$, the well-behaved structures which we can actually work with are the pathologies and the rare instances. This is throughout most of mathematics. It's far less likely that a function from $\Bbb R$ to $\Bbb R$ is continuous, differentiable, continuously differentiable, and so on and so forth. And yet, we work so much with these properties. Why? Because those are well-behaved properties, and we can control these constructions and prove interesting things about them. Compact spaces, being "pseudo-finite" in their nature are also well-behaved and we can prove interesting things about them. So they end up being useful for that reason.

Compactness does for continuous functions what finiteness does for functions in general.

If a set $A$ is finite, then every function $f:A\to \mathbb R$ has a max and a min, and every function $f:A\to\mathbb R^n$ is bounded. If $A$ is compact, then every continuous function from $A$ to $\mathbb R$ has a max and a min and every continuous function from $A$ to $\mathbb R^n$ is bounded.

If $A$ is finite, then every sequence of members of $A$ has a sub-sequence that is eventually constant, and "eventually constant" is the only sort of convergence you can talk about without talking about a topology on the set. If $A$ is compact, then every sequence of members of $A$ has a convergent subsequence.

Compactness is the next best thing to finiteness.

Think about it this way:

Let $A$ be a finite set, let $f: A \to \mathbb{R}$ be a function. Then $f$ is trivially bounded.

Now let $X$ be a compact set, set $f: X \to \mathbb{R}$ be a continuous function. Then $f$ is also bounded...