What is the intuition behind the solution of Hilbert's third problem?

I have read a book called "Proofs From The Book", but it defined many terms and contains much terminology, so I couldn't understand how to obtain a proof by using Bricard's condition. However, I couldn't understand the proof of Bricard's condition either, so I have no hope to understand the formal proof, hope you could give me the intuition behind the proof.

Bricard proved a special case of scissors congruence: Any two polyhedra that are mirror images of one another are scissors congruent. And along the way, he discovered a precursor to the Dehn invariant. There is intuition behind the Dehn invariant, essentially relying on a distinction between dihedral angles that are rational multiples of $\pi$ and those that are not. The cube is not scissors congruent to the regular tetraheron basically because the latter has irrational (multiples of $\pi$) dihedral angles while the cube has rational dihedral angles, $\frac{1}{2}\pi$. Of course, there is much more, but this is somehow the essence.