I often get asked this question, but it's hard to answer because our brains aren't well-equipped for imagining more than three dimensions.
Sometimes, people imagine the expanding Universe as the surface of a balloon. Galaxies and stars are stuck to the surface while the balloon expands. The surface of the balloon is curved, and we can tell it's curved because we're looking at it from outside, in our three-dimensional space.
So if our Universe is expanding, does that mean there's some higher dimension it's expanding into?
This is where things get weird. Imagine a line of army ants on the surface of the balloon. Army ants march in straight lines. They try to do this on the surface of the balloon, but the surface is curved, so the line wraps its way all over the surface.
The balloon could even be crinkly in crazy ways so the line of ants even crosses itself in places. The ants don't understand this because they are so tiny that they don't know the surface is curved - like flat-Earthers they think their land is flat.
It turns out that the only thing you need to completely describe the curviness of space in relativity is these army-ant lines. Sure, you could describe the curvature of the balloon in our three-dimensional space, but the army ants would do a perfectly good job just looking at the lines they made.
This is handy for us, in our expanding Universe, because we only need measurements inside our own space to measure how it curves. We don't need some gigantic cosmic surveyor living in a higher dimension to do it for us. The mathematical equations that describe the curvature only use measurements we make ourselves.
But then we hit a philosophical problem. If we can measure and define everything we need from inside our curvy space, there is no need for it to live in a higher-dimensional space. Maybe there is a higher dimension out there, maybe there isn't. But there is no need to suppose it's there.
Now, I've been talking about 'space' this whole time, but in Einstein's relativity, time and space aren't separate. Instead of a three-dimensional space and a single time dimension, we have a four-dimensional composite called spacetime. Personally, I find three dimensions hard enough to imagine, and a curved three-dimensional space is next to impossible. A four-dimensional spacetime is pretty much impossible for me, except in a few limited examples, and as for curved spacetime - no way. Human brains haven't evolved the cognitive machinery for doing this. I genuinely believe that physics makes the hardest demands on a person's imagination of any subject.
It turns out that the way that curvy space gets described can also be used for spacetime. When you do this though, you find you can also describe the expansion or contraction of the Universe. The expansion or contraction is a sort of 'curvature', but it involves time as well.
And then we hit the same philosophical problem. Like the army ants on the balloon, we can define and completely describe the expansion of the Universe using only measurements from inside our Universe. We don't need some great cosmic giant standing outside to tell us how much the Universe is expanding. Maybe there is a higher dimension out there that our Universe is expanding into, or maybe there isn't. The point is, we aren't required to suppose it's out there.
This is very hard for our brains to imagine: a surface can be curved, but not embedded in a higher-dimensional flat space. And the best answer I can give to "what is the Universe expanding into?" is that our Universe isn't necessarily expanding into anything - it's just expanding.