Here's a quick and simple explanation of the Sum over Histories approach:
Suppose that you had a light source at A and a detector at B. You know that the emitted photon starts at A and ends up at B, but you know nothing else about the path it followed.
Under Feynman's Sum over Histories formulation, you must assume that the photon simultaneously followed ALL POSSIBLE PATHS from the source to the detector. It went directly from A to B via the straight line route... but it ALSO went by bouncing off the left wall, AND it went by bouncing off the right wall, AND it went by going up your pant-leg and out your sleeve, then looping three times around the table, and THEN into the detector...
Sounds silly, right? Everybody knows that light travels in straight lines, and that when it reflects off something the angle of reflection is the same as the angle of incidence. So what's all this nonsense about going "up your pant-leg" and "looping three times around the table"?
But it turns out that if you calculate the amplitude and phase (the up-down vibration) of the photon along every possible path that it MIGHT follow, and sum all of those phase/amplitude results together, you get EXACTLY the result that is actually measured. This bizarre "trace out every possible path and add them up" method predicts precisely the result that you actually measure in your experiment.
As you work through the math, it becomes clear that "light travelling in straight lines" and "the angle of reflection being equal to the angle of incidence" are actually EMERGENT PROPERTIES of the system, not axioms.
And it obviously explains the double-slit experiment... "every possible path" not only includes photons passing through the right slit, and photons passing through the left slit... but because even curved paths are permissible it also includes photons that travelled on a wiggly path and passed through BOTH slits.
It also explains the scattering pattern that you get when you reflect a beam of light off a diffraction grating (something that MWI doesn't explain at all).
The drawback of SoH is that you have to give up the idea that objects (at least on the quantum scale, such as photons and electrons) follow specific paths or trajectories. In fact, the terms "trajectory" and even "location" stops making sense at that scale.
I heartily recommend reading QED by Feynman. You won't have any trouble understanding what he's saying. You might have trouble believing that he's serious, though...
And that's why these are simply interpretations. From what have read so far from that website you posted, you are right about that.