Lévy matrices are symmetric random matrices whose entry distributions lie in the domain of attraction of an $α$-stable law. For $α< 1$, predictions from the physics literature suggest that high-dimensional Lévy matrices should display the following phase transition at a point $E_{\mathrm{mob}}$. Eigenvectors corresponding to eigenvalues in $(-E_{\mathrm{mob}},E_{\mathrm{mob}})$ should be delocalized, while eigenvectors corresponding to eigenvalues outside of this interval should be localized. Further, $E_{\mathrm{mob}}$ is given by the (presumably unique) positive solution to $λ(E,α) =1$, where $λ$ is an explicit function of $E$ and $α$. We prove the following results about high-dimensional Lévy matrices. (1) If $λ(E,α) > 1$ then eigenvectors with eigenvalues near $E$ are delocalized. (2) If $E$ is in the connected components of the set $\big\{ x : λ(x,α) < 1 \big\}$ containing $\pm \infty$, then eigenvectors with eigenvalues near $E$ are localized. (3) For $α$ sufficiently near $0$ or $1$, there is a unique positive solution $E = E_{\mathrm{mob}}$ to $λ(E,α) = 1$, demonstrating the existence of a (unique) phase transition. (a) If $α$ is close to $0$, then $E_{\mathrm{mob}}$ scales approximately as $|\log α|^{-2/α}$. (b) If $α$ is close to $1$, then $E_{\mathrm{mob}}$ scales as $(1-α)^{-1}$. Our proofs proceed through an analysis of the local weak limit of a Lévy matrix, given by a certain infinite-dimensional, heavy-tailed operator on the Poisson weighted infinite tree.