The horn button is a simple switch to the horn.
The output of the switch is still 12v not technically GND.
When you start the car 12v goes to the switch and stops, it also goes to the LED's where it powers them but because they have a small resistance they stop enough power going through to activate the relay.
However when you press the horn button enough power goes through to activate the relay, but when you let go the small amount of power that gets through the LEDs is enough to keep the relay closed.
Basically the relay needs just a little extra push to switch than to keep it made.
I'm sure oldspark will be along to give you a more technical explanation at some point. (and probably debunk what I have just said)