Shut down the power at the house circuit box.
Use your screwdriver to remove the screws that hold the whole existing fixture in place around the base of the fixture (not just the socket). Disconnect the wires behind the fixture and remove the fixture completely.
Hold the mounting plate that came with your new fixture against the electrical box. Secure it there with the provided screws. The exposed wires in the box should still be accessible through the bracket.
Twist the bare ground wire from the electrical box around the green ground screw that's on the mounting bracket. Tighten the ground screw.
Connect the black wire from the fixture to the black wire that's in the electrical box, by twisting a wire cap over the two exposed ends. Repeat for the white wires.
Press the fixture to the bracket, pushing the wiring into the electrical box. Secure the fixture to the bracket with the screws that came with it.
Turn the electricity back on.