It's simple: there are only two elementary actions that can happen to a photon - it is generated at one point and then it is consumed at another. Its whole life consists of just these two events, two "instants" if you will - the "travel" is really just the delay between those two occurrences with regard to everyone else. Two instants, each of zero duration, comprise zero time: $0 \cdot 2 = 0$.
Because of this, it doesn't have the capability to define an "experience" that is anything more than those two events - and thus you cannot ascribe to it a "view" of the Universe. This is why, in a sense, you cannot define a reference frame for a photon that "makes sense", as you're running into. The "world view" is "I get emitted, I get absorbed". Boom. That's it.
Another thing that is often missed is that "reference frames" do not exist as an entity in their own apart from spacetime and its contents. What they are, instead, are just systems of coordinates, that is, ways of labelling events i.e. points in spacetime, and in that case are wholly arbitrary. The Lorentz frames in special relativity are arbitrary - what is not is the Lorentz symmetries that relate them, which are self-maps of the points on spacetime as points regardless of labels, and the fact that the dynamics, the physics, on that spacetime, respect those symmetries, meaning that if we want to describe a moving observer's view of the world, we can do so without altering the form of the equations we were already using - which is how you can talk hypothetically of a Lorentz symmetry violation without it tripping over itself in a contradiction. (Note that you can always create a "moving reference frame" in any sort of universe that has a notion of space and time, but it may be that you need to use different "physics" to talk about this: toy example is Conway's Game of Life. Its rules are not motionally symmetric in any way.)
And to link that back to the idea of "experience", any "way of experiencing" spacetime can be related to some coordinate system. In fact, the Lorentz frame is not any human's experience of spacetime: instead, what you want is a "light cone frame" where the "present" is taken as one's past light cone. The Lorentz symmetry still relates it to the moving case, but the coordinates are all different and the symmetry takes a different form. (Whose experience is it? Well, because its "present" involves knowledge of spacelike-separated events, a patch of it can only be seen by an observer well into the future of the happenings in question, that has reconstructed the view from suitable observations.)
Hence, the failure of the Lorentz transformation to meaningfully define a frame means that the situation in the case of a photon is fundamentally different from the situation in case of everyone else, and its experience is likewise different.
