There are two ways to hook up a relay. Normally closed and normally open.
This link below has a really good diagram on how a relay works.
http://www.bcae1.com/images/swfs/relay.swf
In this diagram they are using the switch to supply power. Now if you take the terminal 30 and hook that to a ground, then the relay is supplying a ground to your horns. When you turn your upfitter switches on (i.e. apply power to terminal 85) you would be switching ground from terminal 87a to 87.
According to the Hadley-Products link you posted before step 2 reads as following:
As shown in figure 3 mount toggle switch (item#36) on or under the dash in a location easily accessible to the driver. Connect wire (item#36) from vehicle ground to either solenoid terminal (item#15). Connect other solenoid terminal to either terminal on the toggle switch. Connect remaining toggle switch terminal in line with the existing horn wire using a wire splice connector (item#38). With toggle switch in “off” position, only the vehicles electric horn will operate. With switch in “on” position, air horns and electric horn will be operated.
Ok, so basically what you are doing is adding a ground switch (as you mentioned earlier) which is acting like an interrupter for the circuit.
When you go to hook up your horns, you will do the following:
Connect wire (item#36) from vehicle ground to either solenoid terminal (item#15). Connect other solenoid terminal to terminal 87 on the relay. Connect the terminal 30 on the relay to the existing horn wire using the wire splice connector (item#38). Then you will need to go and purchase some additional wire for the relay. Hook terminal 86 on the relay to the ground. Then run your power from your upfitter switch to terminal 85.
What this does it the following:
When the up fitter switch is off, there will be no power to the relay and the ground signal that is being fed to the terminal 30 will be sending it to the 87a terminal, which in this case is dead. When you turn on your up fitter switch, you will energize the relay, which will connect terminal 30 to terminal 87. This will send the ground connection to the solenoid.
Does any of this make since?
If so, you have two options for hooking up the relay, you can leave it so the air horns always work with your regular horn, and when you flip the up fitter switches on, the air horns won’t work. (if you want it to work this way, then use the terminal 87a instead of 87)
Or you can wire it like I mentioned above and they will only work when the up fitter switch is on.