We study the random connection model on hyperbolic space $\mathbb{H}^d$ in dimension $d=2,3$. Vertices of the spatial random graph are given as a Poisson point process with intensity $λ>0$. Upon variation of $λ$ there is a percolation phase transition: there exists a critical value $λ_c>0$ such that for $λ<λ_c$ all clusters are finite, but infinite clusters exist for $λ>λ_c$. We identify certain critical exponents that characterize the clusters at (and near) $λ_c$, and show that they agree with the mean-field values for percolation. We derive the exponents through isoperimetric properties of critical percolation clusters rather than via a calculation of the triangle diagram.