It’s slightly more complicated than this but it’s due to quantum mechanics. Specifically, it’s due to limitations on the allowed energy levels of electrons. Electrons in an atom actually exist as probabilistic clouds where at any point there is a non-zero probability of finding the electron when you measure it. The shapes of these clouds – the wavefunctions of the electrons – are described by the Schrodinger Equation, and different energy levels have different shapes. Additionally, energy is quantised; the minimum resolution of energy levels is the Planck energy.
For an electron to fall into the nucleus it would have to emit energy to go down one energy level. However, electrons cannot fall to lower energy levels than their ground state due to Heisenberg’s uncertainty principal and the quantisation of energy.
A common, though flawed, way of thinking about it is as a standing wave around the nucleus. A standing wave must have an integer number of periods, and the electron can only exist as valid forms of these standing waves. If the electron is in a state corresponding to the standing wave with the minimum number of periods, then it doesn’t make sense for it to move to a lower level.
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