I will ask my question in a way that's all handwaving and no math, and I will welcome handwaving answers. I'm interested in visualizing the concepts.
We had an idea about the atom as a massive nucleus with positive charge, surrounded by lightweight negative charges. But we knew that opposite charges attract. So why didn't the electrons fall into the nucleus?
We had a model for that. Masses attract, and the earth doesn't fall into the sun because it's in orbit. Maybe the electrons were in orbit around the nucleus.
But accelerated charges always radiate. Accelerated electrons would radiate away their energy. They would fall into lower and lower orbits until they fell into the nucleus. So that model failed.
At that point there were various alternatives available.
We could figure that charges don't always attract, and look for a reason that the attraction didn't happen in this specific case.
We could figure that accelerated charges don't always radiate, and look for a reason that these particular charges don't radiate.
We could postulate a hidden repulsive force, a sixth force, that balances away the charge attraction in this particular case. Why not? We did that for the strong force and the weak force. Why not one more?
Or we could use the following reasoning: Since accelerated electrons always radiate, electrons must be mostly stationary. They continually absorb random radiation from elsewhere and radiate it away again. So they jostle around, but they don't have any consistent velocity and they give as good as they get. They are mostly stuck in place near specific areas, and we can statistically estimate where those areas are. It takes a set quantum of energy to get from one of those areas to another (or the energy is lost as radiation to go the other direction.) Those are the only places electrons can be close to a nucleus, so they can't fall in.
Free electrons don't have to emit quantized radiation. Electrons in a radio tower can emit any frequency the tower operator chooses. But electrons in an atom do.
What is it about an atom that keeps the electrons from falling into the nucleus? Why are they stuck in quantized orbitals? Opposite charges attract, that's the biggest fundamental force, what is the opposing force that keeps it from happening this particular time?
