Why is light invisible?

Because Maxwell's equations are linear. Equivalently there is no elementary photon-photon interaction. If there were, say, a quartic photon interaction then you would be able to see a beam of light directly instead of seeing its interaction with dust particles.

The other question is sort of a duplicate but only if the OP understands why. (Or it's actually what he was asking, which I'm not sure it is.)

When you "see" something, it's because your brain is interpreting the interaction of photons with material in your eye. Your interpret more interactions as brighter and different energies as different colours. The only time you see photons, therefore, is when they travel into your eye. So if they're just going past or buzzing around, there's no reason you would see them. To see something, light must scatter off it. The way in which it does so determines how the object looks.

The other question asks about photon self-interaction. Its related because, if photons interacted with themselves, then photons would scatter photons into your eye, and you'd see them.

This may be seem like an irrelevant tangent but I hope this helped answer your question...

According to common usage, one doesn't see the photons reaching the eye.

Instead we say that we see the objects who emitted them in the direction of our eyes. To some extent we also see objects in a light background if they absorb the background light.

This means that to see an object it must either be able to emit/reflect photons or absorb enough of them. Neither is the case for light itself, as (due to the linearity of the Maxwell equations) it doesn't interact with itself. (Actually there are tiny quantum corrections that lead to very weak interaction, but this is by far not enough to be easily observable.)

However we see beams of light in a dusty room, as photons scatter off the dust and so the path a beam travels is illuminated. Note that we don't say that we see a beam of dust, though strictly speaking this is what reflects the light and hence is seen.