This study presents a novel numerical model for laser ablation and laser damage in glass including beam propagation and nonlinear absorption of multiple incident ultrashort laser pulses. The laser ablation and damage in the glass cutting process with a picosecond pulsed laser was studied. The numerical results were in good agreement with our experimental observations, thereby revealing the damage mechanism induced by laser ablation. Beam propagation effects such as interference, diffraction and refraction, play a major role in the evolution of the crater structure and the damage region. There are three different damage regions, a thin layer and two different kinds of spikes. Moreover, the electronic damage mechanism was verified and distinguished from heat modification using the experimental results with different pulse spatial overlaps.