The motivation for this construction is explained here:
He first gives the example of a space of constant positive curvature - the 3-sphere, given by taking a flat Euclidean space of one higher dimension (4) and restricting to the subspace $(x_1, x_2, x_3, x_4)$ s.t. $$x_1^2+x_2^2+x_3^2+x_4^2=a^2$$ for some $a$. The metric on the sphere is just the induced metric from the embedding.
If, instead, we start in Minkowski space, and look at the subspace $(x_0, x_1, x_2, x_3)$ s.t. $$x_0^2-x_1^2-x_2^2-x_3^2=a^2$$ then we end up with a space of constant negative curvature, which we can draw as one sheet of a hyperboloid. Again the metric is just the induced metric.
De Sitter space is just exactly the same thing but starting in 5 dimensional flat Minkowski space instead of 4.
Your interpretation is right, but it's important to notice that time as defined here (the one in which the spatial 3-sphere contracts and expands) isn't a Killing vector of the spacetime, i.e. isn't a symmetry like in Minkowski space.
This post imported from StackExchange Physics at 2014-03-22 17:08 (UCT), posted by SE-user twistor59