But it actually is! It's just written in an excessively jargon-laden manner. Here's an explanation that's less jargon-laden:
We can describe a polynomial by its coefficients, standard x^5 + ax^4 + bx^3 + cx^2 + dx + e = 0 form. Actually, we're going to restrict a, b, c, d, and e so that they are all determined by a single parameter B (this is the Brioschi family of quintics). Remember that all values here are complex numbers, not real numbers.
Now what happens if we move around B a little bit (https://duetosymmetry.com/tool/polynomial-roots-toy/ is very helpful for seeing what I mean here)? Well, the roots will change a little bit. There's a continuous mapping going on. Now move around B in a large, closed path that includes a specific point in the interior. It turns out that the continuous motion of the roots will result in them swapping places--a permutation. This is the "monodromy map".
Now it turns out that the set of possible permutations has implications on whether or not you can solve it with polynomials. The paper here doesn't go into details, since it's written for a high-math audience (this is the "A_5 is not solvable since it is simple" bit).
If the terms: fundamental group, covering space etc are jargon then more so is the case for words like 'explanation', 'laden' and well, 'jargon' itself. They have precise definitions meant to capture mathematical concepts and ideas for which there can be no adequate words in natural language.
Composing radicals can only express certain types of permutation groups, and it's not too hard to see how it works. The quadratic formula term `-b +- sqrt(a^2 - 4ac)` is basically exploiting the permutation created by `sqrt(b)` which swaps between `+sqrt(b)` and `-sqrt(b)`. A cube root `cbrt(b)` rotates between three values, which are separated by roots of unity `e^(2pi/3 i)`. A fourth root swaps between four values, which are separated by the fourth roots of unity `(1, i, -1, -i)`. Etc.
So the basic "building block" is "rotate 2 points", "rotate 3 points", "rotate 4 points", etc. Even those doesn't express very much: a generic cubic has symmetry S_3, which is arbitrary permutations of three roots, and even that can't be expressed with just basic sums of these, because the only way to get three-fold symmetry is to use a single cube root.
But take a look at the cubic formula: https://en.wikipedia.org/wiki/Cubic_equation#General_cubic_f.... Turns out if you nest them you can create more interesting symmetries. The cubic formula structure has the structure "sums of cube roots of square roots", which give six total values but really it makes three because of cancellation.
I interpret this to mean that as you try to make the larger permutation groups you're increasingly stressing the expressive powers of radicals. The quadratic did fine with two: the quadratic formula is basically "take the midpoint of the roots -b/2a and add the offset between the two `sqrt((b/2a)^2 - c)` with a plus or minus sign". The cubic formula gets messy because the roots don't have to be evenly spaced so you need a way to express that too. The quartic formula is insane (https://www.curtisbright.com/quartic/quartic-png.html) and involves sums of square roots of sums of third roots of square roots. So it is not too surprising that things fall apart soon after, apparently at the quintic.
The specific way that the quintic fails is that the symmetry group S_5 is not "solvable", which is hard to explain without a lot more space, but which basically means that it includes a list of operations (P, Q, ...) which when you replace them with their commutators (PQP^-1 Q^-1, ...) don't get any simpler. I guess that is what radicals don't do: commuting radicals always makes them simpler, and so at N=5 you hit a group they can't fake.
We can describe a polynomial by its coefficients, standard x^5 + ax^4 + bx^3 + cx^2 + dx + e = 0 form. Actually, we're going to restrict a, b, c, d, and e so that they are all determined by a single parameter B (this is the Brioschi family of quintics). Remember that all values here are complex numbers, not real numbers.
Now what happens if we move around B a little bit (https://duetosymmetry.com/tool/polynomial-roots-toy/ is very helpful for seeing what I mean here)? Well, the roots will change a little bit. There's a continuous mapping going on. Now move around B in a large, closed path that includes a specific point in the interior. It turns out that the continuous motion of the roots will result in them swapping places--a permutation. This is the "monodromy map".
Now it turns out that the set of possible permutations has implications on whether or not you can solve it with polynomials. The paper here doesn't go into details, since it's written for a high-math audience (this is the "A_5 is not solvable since it is simple" bit).