dude, that thing in red isn't a button. It's a spring loaded contact that crazy86 explained the workings of. Check to see that when the adapter goes onto the steering shaft, it has a little metal ring on it that touches that "button" you circled. When I changed the wheel on my car the dopey bastard who had the wheel before me stuffed the horn contact ring on the adapter, broke the wire off and stuck it back on with glue. Needless to say the wire didn't make very good contact with the ring thus my horn didn't work.
There should also be 2 wires on that horn button that goes in the middle of the wheel, one goes to somewhere on the adapter, or perhaps the adapter has a wire on it? Need more pics of it. And the other goes to one of the screws that holds the steering wheel to the adapter plate. That's how mine worked anyway.
Do as crazy86 says, touch that wire to anything metal and if the horn still doesn't go check that:
you actually have a horn, the bit that makes all the noise
the fuse isn't blown if there's a fuse for it
the horn thing itself isn't rooted
edit: connect these two bits together with any old bit of wire you can find.
how do I attach a picture? when I hit preview the picture doesn't appear.