There are many applications of the Langlands program to number theory; this is why so many top-level researchers in number theory are focusing their attention on it.
One such application (proved six or so years ago by Clozel, Harris, and Taylor) is the Sato--Tate conjecture, which describes rather precisely the deviation of the number of mod $p$ points
on a fixed elliptic curve $E$, as the prime $p$ varies, from the "expected value" of $1 + p$.
Further progress in the Langlands program would give rise to analogous distribution results for other Diophantine equations. (The key input is the
analytic properties of the $L$-functions mentioned in Jeremy's answer.)
At a slightly more abstract level, one can think of the Langlands program as providing a classification of Diophantine equations in terms of automorphic forms.
At a more concrete level, it is anticipated that such a classification will be a crucial input to the problem of developing general results on solving Diophantine equations. (E.g. all results in the direction of the Birch--Swinnerton-Dyer conjecture take as input the modularity of the elliptic curve under investigation.)