This was addressed in considerable detail in the film's official novelisation. In short, the artificial gravity is on the fritz and the gravity generator isn't getting enough power. This causes intermittent failures on various decks as the gravity switched between local (ship's) gravity and the real gravity from the planet below.
Scott grabbed for a handhold as his feet momentarily left the floor and he started to slide up the near wall. The ship’s gravity precessers were beginning to struggle against the competing pull of the Earth’s own gravity.
“There’s not gonna be an evacuation if there’s no power to stabilize the damn ship! ” he shouted at the captain. “We lose power to the precessers and artificial gravity will be all over the place. People will be literally climbin’ the walls trying to get to their evac shuttles.”
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Throughout the confusion, it was all the ship’s compromised computer system could do to keep the artificial gravity on the wounded Enterprise from slewing crazily from one degree to another and slamming its crew around like ball bearings in a barrel.
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The difficulty arose from the conflict between Earth’s intensifying pull and the increasingly erratic operation of the ship’s artificial gravity system. While the crew was prepared to deal with an emergency that saw them walking on floors one minute and ceilings the next, the constant gravitational flux forced everyone to go very slowly to avoid injury. As a result, the majority of the crew had yet to make it halfway to their designated shuttles.
