The universe doesn’t have to be expanding into anything in order to expand. I know that sounds ridiculous, so let me give you a different example that is easier to understand.
Imagine that you have a line that goes on forever. On that line, you have a mark every inch. There are an infinite number of inches. Now move each marker so they are separated by two inches. The whole pattern has expanded. It still goes to infinity, but the markers are further apart. The pattern has expanded, but the length is still infinite.
Now a new example. Suppose you have a long piece of rubber, going all the way to infinity. (That piece of rubber represents the universe.) The rubber has marks on it every inch. Now stretch the rubber, until the markers are two inches apart. It still goes to infinity — but it has expanded.
Physicists think of “space” not as emptiness, but similar to a piece of rubber. (But they don’t call it rubber; they call it the “vacuum”. “Particles”, in physics, are just vibrations of the vacuum.) The vacuum can expand, just like the piece of rubber. But because it goes all the way to infinity, it doesn’t need more space. A clever way to say it is that “there’s lots of room at infinity”. (That’s clever, but it doesn’t really explain anything.)
Now here is something new that might confuse you, or might help. In the standard physics theory, the galaxies are all getting farther apart; that is the expansion of the Universe. Yet in the way the theory describes it (I mean in General Relativity Theory) none of the galaxies are actually moving. All that is happening is that the amount of space (vacuum) in between them is increasing.
No, you will not learn this in school, or even in college (unless you have an extraordinary professor). It is usually taught in graduate school, when you are earning a Ph.D. degree. At that point the language you will encounter is this: “In the Big Bang Theory, all galaxies have fixed coordinates. (That means they are not moving.) The ‘expansion’ is described by the ‘metric tensor’, which describes the distances between those fixed coordinates. In the Big Bang Theory, it is the metric tensor which is changing; that represents the expansion of the Universe, even though the galaxies aren’t moving. The recent discovery of accelerated expansion means that the rate of expansion is increasing.”
Maybe you’ve read about the curvature of space. Put a black hole between two unmoving objects, and the distance between them will suddenly increase — even though they haven’t moved. So “distance” is not as simple as people thought. It was Einstein who came up with the remarkable idea that “space” (that is, vacuum) is flexible; it can curve and stretch.
I expect you will find this to be very confusing. That’s not a bad sign; it is a good one. When you learn new things that are completely different than you ever imagined, then “confusion” is the first step.