Find the end of the LED strand that has the female connector. This should be the OUT end of the strip. Double check this is right by looking for an arrow on the back of the LEDs pointing in the direction of data flow.
Use a pair of wire cutters to cut this connector off with a few inches of wire still attached.
Separate the three wires on the connector and strip about 1/4" from each one using a pair of wire strippers.
Wrap the bare ends of the wires through the holes on the Gemma as shown, and use a soldering iron to solder them in place. The red wire will go to VOUT, the middle wire goes to D1, and the remaining wire goes to G.
If you're using a LiPoly battery, take a rubber band and carefully wrap it around the battery, trapping the wires inside. This will provide strain relief so your battery leads don't break off. This is a weakness of LiPoly batteries, and the rubber band trick will save you lots of heartache.
Plug your LiPoly battery or your AAA battery holder into your Gemma and see if the onboard light comes on. If it doesn't, try flipping the tiny on/off switch on the face of the Gemma. If it still doesn't, be sure your AAA battery holder's on/off switch is also turned on and that your batteries are charged.