For roughly two hours a night, you are almost completely unable to move. You've never noticed, because you were busy.
While you dream, your brain sends the same movement commands it sends when you're awake. You're running, you're reaching, you're getting out of the way of something. Those signals are real and they're on their way to your muscles.
So something has to stop them.
A gate closes at your neck
During dreaming sleep your body switches almost all of your muscles off. The commands still leave. They just don't arrive.
The muscles that stay online are the ones you can't afford to lose: your breathing, and your eyes — which is where the rapid eye movement in REM sleep comes from. Everything else is held down.
The reason is obvious once you see it. Without that gate you'd act out everything you dream, in the dark, at speed, near furniture. There are people whose gate doesn't close properly, and they do exactly that.
When the timing slips
The switch doesn't always line up neatly with waking. Come round while it's still on and you get sleep paralysis: awake, aware of the room, and unable to move for a few seconds. Frightening, harmless, and far more common than people admit — a good chunk of people get it at least once.
It's more likely when you're short on sleep, on an irregular schedule, or sleeping on your back. If it happens, it passes on its own. Breathing is one of the things that never switches off, so that part is fine even when it doesn't feel it.
And the dead arm?
That's something else entirely, and much more boring: a nerve squashed against a mattress for too long. Normal, temporary, and nothing to do with the gate.