"It's not expanding "into" anything. Like all of the curved spacetimes we talk about in general relativity, the spacetime describing an expanding universe isn't embedded in some higher-dimensional space. Its curvature is an intrinsic property.
To be specific, it's the property describing how we measure distances in spacetime. Think about the simplest example of a curved space: the surface of a sphere. If I give you the longitudes of two points and tell you they're at the same latitude (same distance from the equator) and I ask you to tell me how far apart they are, can you do it? Not without more information: those two points will be much further separated if they're near the equator than if they're near the North or South Pole. The curvature of this space means that distances are measured differently at different points in space, particularly, at different latitudes.
An expanding universe is also a curved space(time), but in this case the curvature doesn't mean that distances are measured differently at different points in space, but at different points in . The expansion of the Universe means quite simply that the distances we measure between two points which are otherwise stationary grows over time. In effect, the statement that "space" is expanding is really a statement that our cosmic rulers are growing."