The universe is not expanding into anything, according to the theories that describe the expansion. The universe is everything, so it does not require space to exist "outside" it. The way we define space in our universe does not require additional exterior space into which it can expand since an expansion of an infinite expanse can happen without changing the infinite extent of the expanse. Even if the overall spatial extent is infinite and thus the universe cannot get any "larger," we still say that space is expanding because, locally, the characteristic distance between objects is increasing. The universe has no center and no edge, and the Big Bang did not happen in one location in space but everywhere in the cosmos simultaneously. The universe is expanding in the sense that the distances between galaxies, on average, grow with time. The universe is like a balloon that is inflating, and the galaxies are like coins glued onto its surface. As the balloon expands, the coins on the surface of the balloon move away from each other, and if you are living in a galaxy, you will see all the coins moving away from one another. The universe is expanding, but it is not expanding into anything.