We present hydrodynamic simulations of a major merger of disk galaxies, and study the interstellar medium (ISM) dynamics and star formation (SF) properties. High spatial and mass resolutions of 12 pc and 4 × 104 M sun allow us to resolve cold and turbulent gas clouds embedded in a warmer diffuse phase. We compare lower-resolution models, where the multiphase ISM is not resolved and is modeled as a relatively homogeneous and stable medium. While merger-driven bursts of SF are generally attributed to large-scale gas inflows toward the nuclear regions, we show that once a realistic ISM is resolved, the dominant process is actually gas fragmentation into massive and dense clouds and rapid SF therein. As a consequence, SF is more efficient by a factor of up to ~10 and is also somewhat more extended, while the gas density probability distribution function rapidly evolves toward very high densities. We thus propose that the actual mechanism of starburst triggering in galaxy collisions can only be captured at high spatial resolution and when the cooling of gas is modeled down to less than 103 K. Not only does our model reproduce the properties of the Antennae system, but it also explains the "starburst mode" recently revealed in high-redshift mergers compared to quiescent disks.