First, I had to remove the cover plate on the back of the refrigerator. Then I unscrewed the water inlet valve from the rest of the assembly. I then unplugged the electrical contacts by pulling them apart. There are two fittings on the valve. One has a nut on it. That one just needs the nut unscrewed, after which the pex tubing can be pulled out. The other one has a pex connection. This one can be separated by simultaneously pushing in on the fitting neck while pulling the pex tubing out. On this one, you might want to cut off a half inch of the tubing as is is scored from the old connection.
To install the new pump, just push in the tubing all the way and screw down the nut securely on the one fitting, and on the other fitting, just push the pex tubing in all the way. Then plug in the electrical contacts again and screw the assembly back together again. Then put the back plate back on the refrigerator and you are good to go.