The horns themselves usually work by vibrating a metal diaphragm.

The reason that the circuit only works when it is installed is that yes,
it does require a ground to operate.  Generally automotive accessories
only have a +12V feed, the ground is provided through the body/chassis -
unless of course you are working on a Corvette or Avanti or something
like that.

But anyway, generally how the circuit works is that there is a +12V feed
to the horn relay (mounted under the hood somewhere) and the ground side
of the coil is sent up to the steering wheel.  The horn switch in the
steering wheel switches that to actual ground (steering shaft) - that's
why the horn doesn't work unless you actually install the switch.  (the
reason that it is done this way is that the coil will limit the current
in the wire up the column, otherwise if it were switched on the 12V side
there would be a possibility of things melting if there were a short in
the wire in the column.  with the typical connection your horn will
blow, but that's easily solved by unplugging the horn relay.)  when that
connection is made, the relay pulls in and sends 12V to the horns
themselves.  But as I stated above, most automotive accessories require
a chassis ground to operate - so that's why your horns don't work if you
are holding them in your hands.

nate

There is an electromagnet that pulls on a metal diaphragm (various
specific methods exist, just giving the generalities here) inside the
horn. As the diaprhagm moves, a contact opens and interrupts the current
through the electromagnet so that the diaphragm returns to its original
position, thus re-establishing electrical contact. Repeat 400+ times per
second... :-) The vibrating diaphragm is coupled to the outside world
via a tuned-length passage that resonates at the note the horn is
designed to produce.

Almost all systems on a car utilize the chassis as the return (ground)
connection. That reduces the wire count by nearly a factor of 2, saving
weight, complexity, and reducing the number of potential failure points.

A lot of cars have floating grounds now that produce those results.  I
noticed it 1st when I installed an aftermarket stereo in a Nissan Sentra for
a friend of mine.  I always check to make sure it works before I lock them
down in the dash.  Hers wouldn't fire up so I checked & double checked all
the connects.  After then doubting my own expertise, I just slid it in the
dash so I could get a 2nd opinion and it just turned on.  I grabbed it to
check for a shortage, and it went off.  That's when I noticed that the
negative lead was basically a dummy lead.  It had noting to do with the
supply circuit to the Nissan Sentra Stero.  The frame in the dash was the
ground/neg terminal.  I supposed Honda has adopted some of the same.