It isn't actually expanding into anything; there isn't anything for it to expand into. The universe isn't getting bigger - scientists tend to use the image of a balloon being blown up to try and get the general idea across, but the problem with that is that it makes you imagine the universe is actually taking up more and more space, the way an expanding balloon does. But the balloon gets bigger because it is being filled with air, while the universe isn't being filled with anything.
Physicists are still not quite sure whether the universe is infinite, though it seems most likely that it is; but even an infinite universe only means that if you could keep on travelling for long enough you would end up back where you started. And if space is infinite, then it goes on expanding infinitely without any need for extra space to expand into.