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.
There are two primary forces relevant to nuclear stability. Electromagnetic charge, and the nuclear strong force. Electromagnetic force attracts opposite charges and repels similar charges (like pushing the “wrong” sides of two magnets together). The strong force pulls all particles together, but only works on really small distances.
In a large atom the nucleus has lots of protons. They really don’t want to stay together because they’re all positively charged. Neutrons spread out the protons, effectively diluting the EM repulsion, allowing the strong force to keep it together.
A hydrogen atom needs no neutrons at all because it only has a single proton.
Imagine two people working in a group firing arrows in a circle. For one person it would be easy, they can use the others people notches to hold the bow.
However, if two people were firing the arrows together, the one has to put their finger on the others persons arrow to hold it in place. If the second person were to hold the first persons arrow, they would both have to hold both arrows or one would slip out. So for two people to fire the bows together, they each have to hold both bows.
Now, do you see the connection you’re looking for?