This is not an answer to the question, "What is the deeper meaning of Abhyankar's conjecture?" It is an answer to the question, "What is one application of Abhyankar's conjecture?" Over an algebraically closed field $k$ of characteristic $p$, every $k$-action of a finite $p$-group on every proper, separably rationally connected $k$-variety (e.g., every rational $k$-variety) has a $k$-point fixed by the entire group. In particular, every finite $p$-group in every semisimple group of adjoint type is contained in a Borel subgroup (maybe there is a direct proof of this, but I do not know it).
The proof, following an argument introduced by Kollár and Debarre, combines Raynaud's solution to Abhyankar's conjecture with the theorem of de Jong and myself (the positive characteristic generalization of the theorem of Graber, Harris and myself). Please confer the following answer of Chambert-Loir as well as my comment, Are rational varieties simply connected?.
Edit. There are two remarks. First, once the $p$-group is contained in a Borel subgroup, automatically it is contained in the unipotent radical of that Borel subgroup, since the multiplicative quotient has trivial $p$-torsion group (the $p$-torsion group scheme structure, of course, is nontrivial). Second, this result completely fails in characteristic prime to $p$. For every integer $n$ that is divisible by $p$, for every algebraically closed field $k$ of characteristic prime to $p$, there is a copy of $\mathbb{Z}/p\mathbb{Z}\times \mathbb{Z}/p\mathbb{Z}$ in $\textbf{PGL}_{n,k}$ that is contained in no Borel subgroup. There is a lift of this group to a $p$-group of order $p^3$ in $\textbf{SL}_{n,k}$.