ORLANDO, Fla.—Police broke down a fence that enclosed the patio of Pulse around 2:15 a.m. Sunday as Ivory McNeal slipped from his hiding place behind the potted plants. “Please don’t shoot!” he said.

Minutes earlier, a man identified by authorities as Omar Mateen had exchanged gunfire with the uniformed police officer stationed at the club to gain entrance. “All the music stopped immediately,” said Mr. McNeal, who was standing near the main bar. “The gunshots just kept going, kept going…people starting to fall on the ground,...