This is a theorem of Lang's from 1956. Here's an online document giving a proof (in the form $H^1(A,k)=0$):

Lecture 14: Galois Cohomology of Abelian Varieties over Finite Fields, 
William Stein. http://wstein.org/edu/2010/582e/lectures/582e-2010-02-12/582e-2010-02-12.pdf

Stein notes that there is a "more modern proof" in the first few sections of 
Chapter  VI  of  Serre's
*Algebraic  Groups  and  Class  Fields*.

The original article is Serge Lang, "Abelian varieties over finite fields," *Proceedings of the National Academy of Sciences* **41.3** (1955): 174-176. It is available at http://www.pnas.org/content/41/3/174.short. It's very short, but phrased in the "old-style" Weil language of algebraic geometry.

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Here's a quick sketch of a proof with $k=\mathbb F_q$. Choosing a point of $X(\overline{k})$, we can make $X$ into an abelian variety over $\overline{k}$. Then the $q$-power Frobenius map $\phi_q:X\to X$ is the composition of an isogeny and a translation, say $\phi_q(x)=f(x)+x_0$ with $f:X\to X$ an isogeny (defined over $\overline{k}$) and $x_0\in X(\overline{k})$. The fact that $\phi_q$ is purely inseparable implies that $f$ is purely inseparable, and hence $1-f$ is separable (look at the action of $(1-f)^*$ on differentials). Use this to conclude that there is a point $x_1\in X(\overline k)$ satisfying $(1-f)(x_1)=x_0$. This implies that $\phi_q(x_1)=x_1$, and hence that $x_1\in X(k)$.