I think a lot of the confusion for this comes from people picturing all of the matter in the universe scrunched down to one point in space that already exists. Then the big kablooey happens and all of that matter explodes outward into the empty space.
That’s wrong.
The way to look at this is to imagine that you stick a flag at one particular point in space. That flag then never ever moves. It says right where it is, no matter what. Then you go some distance away and place another flag there. This flag also never ever moves and stays exactly where it is.
Then sit back and watch. The two flags slowly move apart from each other. But the flags aren’t moving.
And that is the key to understanding it. The flags themselves don’t move at all. The space in between them stretches.
If you rewind that backwards, then the flags get closer together, because if you reverse time, everything squishes, so much that space itself squishes down to a single point. That point is the “big bang”. It wasn’t a bunch of stuff exploding into space, it was space itself exploding into existence.
Distant galaxies aren’t moving away from us. They are staying pretty much where they are (aside from some local motion). And we are staying pretty much where we are. But the space between us is getting bigger, which makes the galaxies farther and farther away from us even though they aren’t moving.
Take a rubber band and stretch it. At any point on the rubber band, each little part is stretching away from its neighbor by a small amount, but either end of the rubber band is adding up all of those little amounts, so the farther away you are along the rubber band, the farther away you move away from the end of the rubber band.
Distant galaxies aren’t “moving” away faster because they have always moved faster. They are staying where they are. They appear to be moving away faster because there is more space between us and them, which means more stretching. If you have twice as much space, that’s twice as much space to stretch, so the apparent motion is twice as much.
Let’s say you have a balloon. Now you take a magic marker and draw two dots right next to each other. Then you draw a third dot on the other side of the balloon. Blow the balloon up, and the two dots are relatively close to each other, but the third dot is much father away. But any two dots drawn right next to each other anywhere on the balloon won’t move that much.
That is why distant galaxies “move” faster. Space is stretchy.