There is a water pump in the machine which pumps the water out of the outer tub and into the drain, after the wash and rinse cycles. It is possible that this pump isn't working properly. I think if it weren't working at all, you wouldn't be able to get beyond the wash cycle. It has been my experience that small articles of clothing may get washed or spun out of the inner tub and into the area around the pump pick up, thus partially blocking the flow to the pump. It's also possible that the drain hose from the pump to the drain may be partially clogged. It is also possible that the pump doesn't run sufficiently long to get all of the water out because its' motor gets hot and shuts down or because of a problem in the controls mechanism which manages the the starting and run time of the pump.
Firstly, I would unplug the washer from the wall outlet and probe under the inner tub, or possibly remove the tub to see that there is no blockage from clothing. It's also possible that things left in clothing pockets like paper clips, small plastic items, etc. can get into the pump and lock up the impeller. If you're sure it isn't blocked you can then run a "short cycle", no load and water only, and watch to see how much water comes out of the discharge hose where it's hung on the drain pipe. It should gush out, and tub should be free of standing water prior to the start of the first rinse cycle. If it does not, then you'll have to access the pump and clear it of obstructions or replace it, if defective. A fair handy man can get to pump either from the rear of the machine with back plate off, or by removing the cabinet cover from the frame, depending on the model of the the machine... Luck
One time appliance repairman...