The mathematical approach of orbitals would give a reasonably simple explanation to the three p orbitals, but I guess it needs 2-3 years studying after high school.
Would it suffice if I say that orbitals must "differ enough" from an other?
s orbitals can't be oriented because they are spherical, so there is only one of each. They differ only by the way they evolve with the distance to the nucleus. 1s keeps the same sign and decreases steadily with the distance, while 2s passes by 0 and changes its sign once, 3s passes twice by 0 and changes its sign twice, and so on.
2p orbitals change their sign once when you observe them along a circle around the nucleus: one lobe +, the other -. There is only one manner to do this, but the direction of the lobes can be chosen. If you align the lobes along x, y and z, these orbitals are "different enough". There are 3 such directions defining three 2p orbitals, and since electrons can have two values of the spin that make them "different enough" too, p orbitals can host up to 6 electrons while s orbitals can host up to 2 electrons.
Other orbitals: d, f... have more possibilities to vary, so there are more of them.
This "different enough" is supposedly outside high school programme. It's called "orthogonal", but not in the sense of geometric angles. It means that the product of both orbitals, integrated over space, is zero. A consequence is that one orbital changes its sign where the other doesn't, so the regions of positive and negative product compensate. You can check that visually between 1s and 2s, 1s and any 2p, 2s and any 2p, 2p
x and 2p
y... But between two 2p orbitals with angles other than 90° the integral of the product wouldn't be zero.
This is a consequence of the very nature of electrons, which are "fermions". It does not result from a force like the electrostatic repulsion, but is a mathematical necessity for them. Photons, which are "bosons", behave differently: you can put arbitrary numbers of them in the same state, as lasers do.
A different approach gives the same conclusion: orbitals are some solutions of an equation that tells the behaviour of electrons, and these solutions are orthogonal.
A very nice site to see orbitals:
https://winter.group.shef.ac.uk/orbitron/choose in the list at left to see orbital shapes.