SCIENCE TUMBLR, EXPLAIN!
Hello that is a touch lamp and basically it uses something called capacitors to turn off and on. Capacitors in touch lamps store charges and when those charges are altered, they activate an oscillator responsible for turning the lamp off and on.
Humans have what is called body capacitance; we can store tiny electrical charges in our bodies and use them to activate touch sensors (which is why your phone's touch screen works, too, and why it won't work if you're wearing cloth gloves that aren't conductive), for example, altering the charge in a capacitor like the one in this lamp.
The cat doesn't have enough body capacitance to set this lamp off, but it IS conductive enough to relay a human's charge. So when the human makes contact with the conductive bare skin of the cat's nose, their charge travels through the cat to the lamp through their toe bean, activating the lamp. Fur is not conductive, so it won't do it when the human just pets the cat.
This explanation was all very sensible until “toe bean” and I am giggling with pure delight XD