This is precisely why I think people shouldn't reify mathematical entities, like spacetime being a literal fabric that "expands" or "curves." When we say space expands we just mean that things that are traveling in a straight line will have their paths curve away from one another. You can imagine two people in a space ship flying next to each other and neither one of them places any pressure on their pilot joystick to turn the space ship, yet over time they find that their paths diverge from each other anyways. This creates a tendency for everything to move away from everything else.

When Einstein first introduced his theory of gravity, it was clear that gravity causes straight lines to curve towards each other, and if you add up all the localized effects of gravity, on a universal scale, the whole universe would on average eventually come together and collapse.

It was believed at the time that the universe is eternally static, so he found a free parameter in theory called the cosmological constant that creates a universal positive curvature (making paths diverge) which if set to just the right value could balance out the totality of the local negative curvatures (making things converge) and give you a static universe.

However, when physicists actually measured the cosmological constant (it's a free parameter like G, you have to measure it) it turns out to be way larger than is necessary just for a static universe.

In the beginning, the universe was all packed down into a very small area. Then the big bang happened, and space started expanding.

Space isn't like a room, it's more like the inside of a balloon.

We don't know if space is infinite.

We could travel at light speed toward the edge of the observable universe, where the cosmic microwave background radiation from the big bang has reached, but even at light speed we'd never be able to catch up. It's had a couple billion years' head start.

Especially since the expansion of space is accelerating. We don't know why it's doing that. A naive hypothesis would have expected it to fall back together, like a cosmic dust cloud gravitates together.

Sorry to nitpick a bit, dark matter isn't connected to the expansion of space (as far as we're aware) but dark energy is probably what you meant. My answer to your question is at the end.

Full disclosure: While I have studied this, my expertise was in a tangentially related field. However a buddy of mine has a PhD in measuring this stuff so I've got some second hand knowledge.

It's a confusing hurdle for any student of physics to understand that spacetime doesn't exist inside another bigger thing into which it can expand, it just kinda exists on its own. Mathematically we don't even treat the expansion quite like growth, it's a bit easier to understand it as our rulers getting shorter, the labels we give to distances changes over time. Personally I like the analogy of a sheet of grid ruled paper.

If you choose two points and count the number of squares between them, divide that grid into a smaller one and then count them again, the "distance" has gone up. Those squares look smaller to us so it seems like the true distance is the same, but the universe doesn't have an external view to make such comparisons from, all we have are the squares and physics obeys them. The point is you can cut squares up forever without running out of squares to cut up, nothing runs out this way.

In spacetime maths (general relativity aka GR) we usually start by defining distances, and when it comes to the expansion of the universe we literally just have a number in that definition that changes over time.

This kind of "our rulers and clocks are dodgy and unreliable" is unfortunately the backbone of this sort of physics. It's a huge pain in the ass, but it's cool af if you're a huge maths nerd.

How does it expand?

🤷

Anyone who can tell you how dark energy works beyond "it has a negative pressure" is full of it. It's a theoretical idea and has never been observed, we just know that if something with negative pressure existed everywhere then it would cause space to expand. Don't quote me, but it's kinda like the opposite of how a black hole squishes spacetime down into a singularity, dark energy pushes out on everything everywhere all at once. (Couldn't help myself it's a great movie go watch it)

There are a bunch of possible things that fit the bill, it could just be a number in Einstein's field equation, it could be a specific type of quantum field that has a constant value everywhere, hell I've even seen models where it's just caused by black holes existing. It's also possible that Einstein got some stuff wrong and that expansion is just what space do. Either way, I don't think these things require more stuff to be created, it's just stuff that's already there.

If I had to take a mostly uneducated guess, I'd say it's probably just a feature of quantum gravity, for which we have no proven theories. Loop quantum gravity just demands it exists for the theory to even be useful, I'm sure string theory has it's own crazy nonsense to explain it too. If we ever do work this out, I fully expect it's just going to be a thing we have to accept exists without an obvious cause, much like how the universe exists but we have no idea why or why the rules it follows are those specific rules and not some others.

Multidimensional expansion is rough, so let's collapse things down to two dimensions. Imagine a rubber band stretched between two chairs. On the rubber band, there are two ants. The ants both walk at exactly the same speed. If they both walk away, they travel at the same velocity. If they both walk towards you, same thing. If the walk towards each other, they are moving twice as fast relative to each other, but their speed is the same. Now, imagine you start pulling one chair away from the other. The ants still walk at the same speed, but the distance between them grows. If they are walking away, the closer one will appear to be moving slower than the one further away.

So with astronomy, we have ants to reference. Light travels at a constant speed. We can also tell if things are moving towards or away from us because of red and blue shifting of light. By measuring the color of the light we see, we can determine it's relative speed, and what we observe is that light from further away is moving away from us faster than light closer. Because light only travels at c, we can only conclude that space is expanding.

There are several other scientific proofs of the concept, but that was the easiest one for me to understand.

As for the Universe being infinite and expanding, let's go back to the rubber band. The band is a circle. There's no end, no beginning. The ants could go around and around forever and never reach the end. Now imagine a balloon, and we have two dimensions for the ants to traverse. They can go forward, back, left, or right. It doesn'tatter how long they walk, they will never reach "the end" of the balloon. If the balloon inflates, the whole thing gets bigger, but it's still unending. The trouble is adding third and fourth and fifth dimensions to the metaphor, because that gets really hard to visualize. Spacetime is all expanding, and we can extrapolate backwards to a time when all of spacetime had zero distance between points. Everything existed as a singularity, all of the universe and all of time compressed into zero dimensions. True story, the person who coined the term "The Big Bang" was using the phrase condescendingly as part of a criticism the theory, but every subsequent study of the theory has supported it.

First off, it's not certain that space is infinite, but I'd say it's probable.

But even if it's infinite, infinity has some unusual properties that make this make sense. For this, I'm going to borrow from Hilbert's paradox of the Grand Hotel.

Imagine a hotel with an infinite number of rooms numbered 0, 1, 2, 3... with no end. The room numbers (and rooms) just keep going on forever. And imagine that hotel has no vacancy. That is, an infinite number of guests are already staying there and there are no hotel rooms that are vacant. But then, someone shows up and asks the hotel clerk for a room. The clerk, being a clever fellow, has all the current patrons change rooms to the room numbered one higher than the one they were previously in. (The person in room 0 moves to room 1. The person in room 1 moves to room 2. etc.) That operation is one that can go on forever. (It couldn't go on forever in a hotel with a finite number of rooms, but in an infinite hotel, an infinite number of patrons can move to the next room up and not a single one of the infinite number of patrons will be unable to do so for lack of a room numbered one greater than their previous room.) Then, the clerk books room 0 for the new arrival. But also notice that the number of patrons before the new arrival is the same as the total number of patrons including the new arrival.

Said another way, ∞+1=∞. (Not only that, but ∞+∞=∞. Thinking about the previous thought experiment, if an infinite number of people arrived to a fully occupied infinite hotel, the clerk could have all the existing patrons move to the room that was double their previous room number and then book all of the infinite number of new arrivals in all the odd-numbered rooms.)

Final thoughts:

• I don't know where you got "the cause of [space expanding?] is apparently dark matter being created." Maybe I'm just uninformed, but I haven't heard of that. (There's the idea of "dark energy" that IIRC is related to space expanding, but I'm not sure I've heard dark matter used as an explanation of that.)
• Some of your question kindof implies that "space" is expanding "into" some... meta-space or something. Like you're envisioning our space existing inside another space. And part of your question is about how the "meta-spacetime" can expand infinitely to accommodate our space. I don't think that's certain. It's entirely possible that's not really an accurate way to view what might be "outside" our space. (I'm not sure "outside our space" is really a meaningful concept.)

Here’s one explanation:

https://www.space.com/space-exploration/james-webb-space-telescope/is-our-universe-trapped-inside-a-black-hole-this-james-webb-space-telescope-discovery-might-blow-your-mind