Another way to think about this that builds some intuition is considering self-interference in the classic double-slit experiment. Usually, this experiment is used to demonstrate wave/particle duality, but that won’t exactly concern us right now.
Think of light being shined (that sounds awkward!) on a plate, with two narrow open slits. On a screen behind that plate, you will observe an interference pattern of lighter and darker areas – thinking of light as a wave, this can be easily explained, like so.
However, if you position detectors at each slit, what you will find is something different – far from observing a continuous variation, each detector will register sharply defined ‘packets of energy’, i.e. light particles – photons (and sure enough, if you do this observation, the interference pattern will disappear, being replaced by just two narrow bands behind the slits, just as you’d get if you’d thrown baseballs instead of photons). Still, it seems in principle possible to account for the interference pattern with this model: a photon travelling through one path in some way interferes with a photon that has taken the other path. This would be the ‘classical’ expectation; photons may do weird things when they’re not being observed, but each of them takes a definite path – it’s just our ignorance, or some sort of loss or hiding of information, that makes us unable to describe them in such a definite way.
However, what you do then is that you attenuate the intensity of your light source up to such a level that you can be sure that only individual photons are being emitted, one at a time. The classical expectation would tell you that no interference pattern should build up – there’s only one photon present at any one time, so what should it interfere with?
Yet, experimental reality disagrees with this expectation: let the experiment run long enough, and an interference pattern will build up that’s indistinguishable from the original one! The conclusion then must be that there is no definite path that is taken by a photon; it doesn’t go either through slit A or through slit B, but, in a sense, through both at once – it doesn’t take one path, but a superposition of paths (or is, as it travels, in a superposition of the states ‘taking path A’ and ‘taking path B’), and this superposition is irreducible, unlike in the classical case where, before looking, the coin is in an uncertain state only descriptionally, but really has definitely landed either heads or tails.
Well, us seeing ordinary light is also dependent on a quantum process – an incident photon changing the structure of some opsin, inducing an electrochemical response. Is there anything substantially different in the way birds see magnetic fields?