If the universe is has a positive spatial curvature on the largest scale then there is a very natural shape---the 3-sphere---which it can adopt, and this shape has a finite volume. If the universe is flat or negatively curved, on the other hand, then the universe could perhaps be infinite, but it could also have various topologies which allow it to be finite.
It is doubtful whether we puny humans could ever have warrant to claim to know that the universe is infinite. How could we possibly know that? We can not. All the data we ever acquire is consistent with a finite universe. (Any unqualified claims that it is infinite that may be made in popular presentations of cosmology are certainly overblown).
Measurements of the spatial curvature, chiefly from the CMB radiation combined with models of early universe physics (the BAO process), give a value slightly above zero but consistent with zero:
\Omega - 1 = 0.006 \pm 0.020
(This is a parameter that relates to curvature; the universe is flat if $\Omega = 1$).
This suggests that the universe is either spatially flat or it is a 3-sphere with a very large radius of curvature. In the latter case, the most natural interpretation is to say that it is indeed such a 3-sphere and then its overall radius would be considerably larger than the radius of the observable universe, and this is perhaps the origin of the comment by Susskind.