If $X$ is a scheme, the Hilbert scheme of points $X^{[n]}$ parameterizes zero dimensional subschemes of $X$ of degree $n$.

Why do we care about it?

Of course, there are lots of "in subject" reasons, which I summarize by saying that $X^{[n]}$ is maybe the simplest modern moduli space, and as such is an extremely fertile testing ground for ideas in moduli theory. But it is not clear that this would be very convincing to someone who was not already interested in $X^{[n]}$.

The question I am really asking is:

Why would someone who does not study moduli care about $X^{[n]}$?

The main reason I ask is for the sake of having some relevant motivation sections in talks. But an answer to the following version of the question would be extremely valuable as well:

What can someone who knows

~~a lot~~something* about $X^{[n]}$ contribute to other areas of algebraic geometry, or mathematics more generally, or even other subjects?

*reworded in light of the answer of Nakajima