Why do we care about the Hilbert Cube? The Hilbert Cube is defined to be the countable infinite Cartesian products of the interval $[0,1]$ or anything homeomorphic to $[0,1]$. Why do we care about this object?
 A: 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.
A: 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):

