I walked into my laundry area to find the floor full of water. When the washer kicked into the spin cycle again, water was gushing out of the area where the washer drain tube is inserted. I also noticed that it sounded like water was running through my drains in the kitchen and bathroom right off the laundry area. I'm wondering what the problem could be and how I can resolve it.

HELP!!

Usually the problem is that your drains are clogged not accepting w/m high water volume. As result, water backs up and overflows. You will need to snake your drain.

If, after snaking the drain from the roof vent or standpipe, and you still have a problem then consider making your stand pipe a closed system using a compression fitting or other fittings,(see images) to seal off the hose in the standpipe. This will force the discharge on down the line with no backups.
Good luck and thank you for rating my answer. Tom