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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.