... согласно которой существует ещё 20 неизвестных ранее науке элементарных частиц.
He's not saying space has 248 dimensions, he's describing the geometry of a polygon. If you read the paper, he's only invoking 3 spatial dimensions and one time dimension to define our universe.
Let's say you've got a cube, and each corner of the cube represents the properties of a subatomic particle. You can have a total of 8 subatomic particles and you can create a direct line between any point on the cube and any other.
E8 is a 248-dimensional set of lines connecting the points of a 57-dimensional imaginary object. What he has done is merge the E8 "object" with the various subatomic particles and used the remaining unassigned points to predict the features of those particles we have yet to detect. In essence, he's created a math representation of a periodic table of subatomic particles.
People with Ph.D's in mathematics aren't expected to understand the theory. People with Ph.D's in particle physics aren't expected to understand the theory.
Quite frankly, there's a serious audience of around one hundred people on the planet that can actually grasp what he's saying, and they seem to be divided about it and its ramifications.