An $n$-dimensional lattice polytope ${\mathcal Q}_σ$ can be associated to any composition $σ$ of a positive integer $n$, as a special case of constructions due to Pitman--Stanley and Chapoton. The entries of the $h$-vector of $σ$, introduced by Chapoton, enumerate the lattice points in ${\mathcal Q}_σ$ by the number of their nonzero coordinates. Chapoton conjectured that this vector is equal to the $h$-vector of a flag simplicial polytope. This paper proves this conjecture. Moreover, it shows that the gamma-vector associated to the $h$-vector of $σ$ is nonnegative by means of an explicit combinatorial interpretation and confirms certain other conjectures of Chapoton on the lattice point enumeration of composition polytopes. A combinatorial interpretation of their $h^\ast$-polynomials is deduced.