Unplug the lamp.
Remove the harp finial, shade harp, lampshade and light bulb. Put them aside for reinstallation when your repairs are completed. The lamp socket should now be clearly visible and accessable.
Squeeze the socket shroud where the word "Press" is stamped into the metal surface. Wiggle it loose from the crimping in the socket base and slide it up over the socket interior. Set it aside with the other lamp parts you previously removed.
Loosen the two contact screws on the socket interior and remove the lamp wires. Once the socket interior is free, discard it.
Loosen the contact screws on the replacement socket interior. Wrap the end of the black lamp wire around the shank of the brass screw and tighten the screw back down. Then, wrap the end of the white lamp wire around the shank of the white metal screw and tighten it back down.
Retrieve the socket shroud. Squeeze it where the word "Press" is stamped into the metal. Slide it down over the new socket interior, and wiggle it back into the crimping in the socket base. You will feel it snap into place when it is properly secured.
Reinstall the light bulb, shade harp, lampshade and harp finial.
Plug in the lamp. Turn the new switch through all three settings to ensure it is working properly.