Everyday experience tells us that, when something expands, it has to have a medium to expand into, so surely it stands to reason that, if the universe is expanding, then it must be expanding into something?
First things first, many would argue that the term "expanding" is misleading. The stars and galaxies do appear to be rushing away from one another for sure, but it might be more accurate to say the the space between them is stretching, rather than the universe expanding like a balloon. This seems like a petty distinction to make, but it means that the galaxies are not moving through a medium, they are effectively stationary, but the space between them is greater. Clear as mud?
Anyway, enough of the semantics, what is it expanding into? The answer is, it depends.
If the universe is infinite then the question is nonsense. something that goes on forever can't have an edge or anything "outside" of it.
If the universe is finite then the question is more interesting. The thing is about physicists, is that they have a habit of talking about the universe in terms of 4-dimensional spacetime, which is impossible to imagine, and suggest that the 3-dimensional universe is embedded in an expanding into 4th dimension.
Picture it as though you are a 2-dimensional being on the surface of a 3-dimensional balloon that is being inflated. You can't perceive the third dimension, so from your perspective, everything is moving away from everything else along every conceivable plane and, what's more, there is no "edge". Now scale it up a dimension, and you have yourself the finite, unbound universe expanding into 4-dimensional space.
Although, the simple answer is really "we don't know".