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 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?.