The following PDF gives an explanation on page 11. Unfortunately I do not know how to reproduce it here.
http://web.mit.edu/~qchu/Public/TopicsInGF.pdf
In short, I am not sure how the symmetry with 18 and 10 arises and the other less trivial parts of the proof (why for example are we looking for coefficients of x6 and not x18)? I have a feeling I am missing something simple with this approach.
Not that it really matters, but in actual dice the $1$ is opposite the $6$, the $2$ is opposite the $5$, and the $3$ is opposite the $4$. Thus if the dice are thrown onto a glass coffee table, the short gambler below the table and looking up sees $7-k$ whenever the person looking from above sees $k$.
So if from above the dice show $a,b,c,d$, then from below they show $7-a,7-b,7-c,7-d$. Thus from below they show sum $28-(a+b+c+d)$.
It follows with $4$ dice, a sum of $x$ and $28-x$ are always equally likely. More generally, with $d$ dice a sum of $x$ and $7d-x$ are equally likely.