How can light be both a wave and a photon?

It just is. It behaves like both. Exactly why, nobody knows, but it does.

Sometimes in physics you just have to suspend disbelief. The maths works, it DOES behave like that, but the real truth is difficult to describe in normal language. The nearest anyone can get to it is that a photon is a packet of waves. At any rate, after three years of physics at university, that worked for me as a way to get your head round it.

Very easily. It comes in "wave packets", each packet having some characteristics of a particle, but the packet consists of a wave.