The inside of the straps have a mesh-like contact surface. Just put them around your arms or hands and turn the device on.
You might feel a slight tingle where the straps contact your skin (particularly if you put it the straps around your forearm). I usually do. It's not terribly pleasant, but it's easily bearable. I especially feel the sensation if there's any moisture (i.e. sweat) on my arms. You don't need moisture, however. You can, however, use moisture on your arm to see if you can feel the tingle, if you cannot feel any tingling without.
If you want to test whether or not electricity is flowing through the device because you cannot feel it, you can get yourself a voltmeter (or multimeter), two alligator clips, and a small resistor (the rating of the resistor doesn't really matter -- most any will do) from Radio Shack. Attach the resistor to each of the wrist straps using the alligator clips. Set the multimeter or voltmeter to measure AC voltage in the 10V range and touch each of the probes to either side of the resistor (i.e. one probe on one side, one on the other). You should get a reading (probably 5V, +/-) on the multimeter. If you get no reaction from the meter, then your zapper is faulty.