A long time ago, Gauss discovered that the curvature of a surface depends only on distances and angles within the surface. He called this the Remarkable Theorem. For instance, a sheet of paper can be rolled on various cylinders or cones, or simply stay flat on the table, but its curvature will be in all cases 0. So, Gauss understood that the geometry of the surfaces is independent on how the surface is embedded in the three-dimensional space, or whether it is embedded at all. This is intrinsic geometry, because of course we can study the way an embedded surface stands in the ambient space (extrinsic geometry). Assuming that the surface is populated by some (fictional) two-dimensional people, as in the book Flatland, they can't determine, by measuring distances and angles or anything else for that matter, how the surface they live in is embedded in an ambient space, or if it is embedded at all.
These findings were generalized by Riemann to any number of dimensions, when he developed Riemannian geometry. A slighty modified version of this geometry, called semi- or pseudo-Riemannian geometry is used to describe spacetime in general relativity.
So, when physicists say that spacetime is curved, or that it expands, they don't mean this with respect to an ambient space in which our spacetime is embedded. They can measure whatever they can in their universe, and they will find no evidence about an environment in which our universe is embedded, something that we would call "outside the universe". So, in general, they will not assume the existence of such an universe.
Some physicists think that there may be an "outside", with more dimensions, of course, and that some physical forces which we observe and even the big bang may be explained by this ambient universe. So far there is no direct evidence of such a thing.