There is a drain line attached to a catch basin somewhere in your refrigerator that is designed to drain any thawing ice or dripping water from condensation away and out of the refrigerator. Sounds like when yours began to melt the ice, it was just too much, the tray filled up. Did it over flow?
Those catch trays under the refrigerator (like in a house), are designed to evaporate as fast as they fill from the heat generated from the refrigerator. When water flow faster than it can evaporate (because you've turned the refrigerator off), it overflows onto the floor.
When defrosting your refrigerator, after a few minutes (leaving the door open), the ice will begin to soften. Using a plastic spachler or plastic pie cutter, pry the ice chunks off the walls and remove them when they fall. Continue to wipe the water from the bottom of the refrigerator inside until everything is thawed and removed.