You'll need a small flat-head screwdriver, and a small phillips head screwdriver (if you're using the bolt-on kit) or a soldering iron (if you're not).
On the side of the power strip is a small green wire connector. Pull that little guy out. Pliers can be your friend if it doesn't slide out easily.
Notice the markings on the box, above the connector. This little diagram tells you which wire is which.
Cut a white and a black wire to length. The length depends on your project -- if you're planning to mount the Circuit Playground someplace far away from the power switch, or use the motion triggers, you'll want to use pretty long wires. If you are planning to use a sound trigger, you can use shorter wires. I made mine the same length as my power cord (about a meter) so I'd have options.
Unscrew both of the little screws all the way. Stick the white wire into the + side and the black wire into the - side. (Labels are printed on the relay box, don't get them mixed up!). Tighten the screws down so the wires stay put.
Stick the connector back in, and double check you've got white on the left, and black on the right. Give the wires a tug so you're sure they're secure.
Strip a bit of shielding from the other end of the wires. Attach the white wire to pin A1 and the black wire to GND on the Circuit Playground. Secure the wires with the bolt-on kit, or solder them in place.
If you're using the bolt-on kit, it's best to thread the wire through the hole first and then insert the screw, rather than winding the wires around the screw head. The wires will stay attached better if you do it this way, rather than the "traditional" around-the-head method.
Note: Do not use pin A0. It is connected to the onboard speaker so it's not so good for switching things. The rest of the analog pins, including A1, work very well.