We develop a first-principles model for the relativistic magnetic reconnection rate in strongly magnetized pair plasmas. By considering the energy budget and required current density near the x-line, we analytically show that in the magnetically dominated relativistic regime, the x-line thermal pressure is significantly lower than the upstream magnetic pressure due to the extreme energy needed to sustain the current density, consistent with kinetic simulations. This causes the upstream magnetic field lines to collapse in, producing the open outflow geometry which enables fast reconnection. The result is important for understanding a wide range of extreme astrophysical environments, where fast reconnection has been evoked to explain observations such as transient flares and nonthermal particle signatures.