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Your problem could be a couple things.
If unplugging or shutting of the unit off stops it filling, then the problem is probably the water level switch.
If the only way to stop it is by turning off the tap, then it is probably the water inlet valve (sticking open)

Water Level Switch
This switch is actually a pressure switch. It has a tube that runs down to the bottom of the tub, as water fills the tub, air is trapped in the tube. As the water rises, the air pressure in the tube increases until it reaches the switch's set point then the switch activates turning off the water and starting the agitate.

Possible problems.
1. Gunk blocking water from entering the tube. Remove the tube at the switch and blow into it to clear the plug.
2. Cracks or holes in the tube allowing the air to escape. Inspect the tube. Also check that it is tight on the switch, if not, cut off half an inch and re-seat it.
3. The switch is toast. Get a piece of tubing with the same internal diameter and attach this to the switch. Start a fill, then blow into the tube. You should be able to stop the water flow without blowing very hard.
If you cannot stop it the switch is toast.

Here is the parts breakdown
Replacement parts for WHIRLPOOL LSR8244DQ0 | AppliancePartsPros.com

Here is the wiring diagram
http://www.servicematters.com/docs/wiring/Wiring%20Sheet%20-%203948172.pdf

No out of the ordinary tools should be required.

OH this was so easy for me.... I went to Washer is Overfilling - How to Repair - YouTube and when I went to my machine. Took out the 2 retaining screws that held the control panel on the top of the machine.... I lifted the panel and there it was.... the clear hose came off... I lightly blew through the hose no restriction. Plugged it back into the water level control switch... selected small load turned it on water flowing started to agitate then I switched to the med load and when it engaged washing I went to large. Now I have a load of laundry washing and no worries. I love YouTube for fixing things. I used it before too when I had a coup-link that broke and I had water flowing onto the floor from under the washer.... I found my parts on e bay and it was cheaper than going to a repair place or buying direct from the manufacturer. Good luck all.