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It is not even that they travel in a pattern that has a wave-like shape. They literally are both waves and particles, and in many ways act like waves. This includes some behaviors that make no sense if you treat them like particles, like how you can fire a beam of one photon at a time through a double slit and somehow it produces a wave interference pattern despite the fact that 1 photon going through at a time could not possibly hit another photon to cause that disruption.
Why exactly does it do this instead of just behaving like a particle? That we don’t have a full and complete explanation for. But it is impossible to deny the reality that that is how it works.
The answer is kinda in the question: is light a wave, or a particle? The answer, given those limits, is that it moves like a wave but acts like a particle when it hits something.
Newton said light is a wave, since you can make it reflect (like a wave) and refract (like a wave).
Einstein got his Nobel not for relativity but for proving that a photon (a particle of light) can “supercharge” an electron in what he called the photoelectric effect.
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Imagine you’re the first astronaut on a distant planet. Everywhere you go there are ducks and beavers, except when you come to a small island in the middle of the ocean and you find a platypus. Is it a duck? Is it a beaver? You can’t answer, because it’s neither of them. It’s a new concept, a platypus, which seems to be part duck and part beaver but is actually something else.
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