So hydrogen can be stable with no neutrons. Helium requires 2 neutrons in order to be most stable, oxygen requires 8, and so on.
So what is it about neutrons that stabilize atoms?
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Protons have *electric charge*. They are pushed away from each other by the electromagnetic interaction.
Protons and neutrons are pulled towards each other by a thing called the *strong interaction*, which is a kind of stickiness.
If you put a bunch of protons near each other they will be pulled together by the strong force, but pushed away from each other by the electromagnetic interaction. With just protons the “electric” push overcomes the “strong” pull and they split apart.
Adding in neutrons helps space the protons out. They will still “electric” push away from each other but not as much (because the “electric” push gets weaker over distance), and the neutrons add in extra “strong” stickiness.
You can think about it like trying to stick two magnets together the wrong way around. You will need really strong glue to stick them directly together. But if you add in a spacer between them (a block of wood or something) you can use weaker glue because the magnetic force won’t be as strong.
As for why we need more neutrons the more protons we have, the “electric” push works over bigger distances than the “strong” pull. If you have a big bundle of protons and neutrons, protons on opposite sides will still be pushing away from each other, but neutrons on the opposite side from our proton won’t be providing much “pull.” Over bigger distances the “electric” push will dominate, so we need to add in even more “strong” stickiness (i.e. neutrons) to space out the protons and to help hold them together.
Eventually adding more neutrons isn’t enough and you get into the atoms that are always unstable.
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