Why are photons massless? ~

"Why" is sort of a deep question. "How we know" is a bit easier to understand. It also depends on what you mean by "mass".

The "why" is U(1) gauge invariance. That's field theory language for the fact that the math we used to describe the interaction between charged point particles called leptons (electrons are an example) is symmetric under certain mathematical transformations. When this happens we have the origin of a physical force at a deep theoretical level.

What falls out of this U(1) gauge invariance is that these leptons interact by exchanging a mediating gauge field. In this case it's a photon. Fields are particles.

So when an electron encounters another electron it interacts by the exchange of photons. The fundamental properties of the electrons don't change. Like their rest mass. Or their spin or charge. Just their energy and momentum. And so the photons have just that. Energy and momentum.

Photons have no rest mass. That gets to the "how we know" part. If they did the behavior of the electromagnetic force over long distance would be different than how we observe it. As would the high energy scattering of photons from leptons. We don't seen that. At a deeper level, aĺl this U(1) gauge symmetry would fall apart and we'd have shite for theoretical physics.

The thing though, photons have no rest mass. If you could slow them down to rest, they have no mass. But they have relativistic mass. That Einstein mass = energy thing. They have effective mass because they have energy. That means they interact gravitationally.

Because they have zero rest energy (extremely short answer)

No interaction with the Higgs field?

Because it makes the math work

How else are they going to travel at light speed?

They are infinitely light.