What is the intuition behind lack of a general solution for 5th and above degree polynomials?

See this detailed sketch of Arnold's proof of Abel-Ruffini theorem here: web.williams.edu/Mathematics/lg5/394/ArnoldQuintic.pdf. It is intended for a layman who can handle some basic group theory (and a bit of topology). It still takes about 5 pages.

Responding to Dmitry Ezhov's comment above: the "finite combination" is necessary, since with an infinite number of operations one can solve an arbitrary quintic. For example, consider $x^5 - x - 1$. Galois theory tells us the roots of this cannot be expressed in terms of a finite combination of radicals and field operations, but using infinitely many it's not too bad: if $x^5 - x - 1 = 0$, then $x^5 = x+1$, so $x = \sqrt[5]{1+x}$. Plugging this back into itself and iterating yields a solution $x = \sqrt[5]{1+\sqrt[5]{1+\cdots}}$ to the original quintic.