Why do we care about the Hilbert Cube?

Every compact metrisable space (of any dimension ) is homeomorphic to a closed subset of the Hilbert cube. Any separable metrisable space is homeomorphic to a subspace of the Hilbert cube. Every separable metrisable topological vector space is homeomorphic to the pseudo-interior $(0,1)^{\Bbb N} \simeq \Bbb R^{\Bbb N}$ of the Hilbert cube.

There is a very nice theory of Z-sets and homeomorphisms of the Hilbert cube (see van Mill’s books on infinite-dimensional topology, or Bessaga and Pelczynski’s book for more on these theorems). It’s a fundamental object in infinite-dimensional topology. The hyperspace of any Peano continuum (in the Hausdorff metric) is homeomorphic to it too, e.g.

In addition to Henno's answer and many comments, here is my take, as a geometric topologist. Below is a quote from the introduction to Chapman's book "Lectures on Hilbert cube manifolds" (Q-manifolds, which are analogues of topological manifolds, except that the local structure is modeled on the Hilbert cube):