In this article, we give a geometric interpretation of the Hitchin component $\mathcal{T}^4(\Sigma) \subset {\rm Rep}(\pi_1(\Sigma), {\rm PSL}_4(\mathbf{R}))$ of a closed oriented surface of genus $g\geq 2$ . We show that representations in $\mathcal{T}^4(\Sigma)$ are precisely the holonomy representations of properly convex foliated projective structures on the unit tangent bundle of $\Sigma$ . From this, we also deduce a geometric description of the Hitchin component $\mathcal{T}(\Sigma, {\rm Sp}_4(\mathbf{R}))$ of representations into the symplectic group