Where do the repeaters actually exist? I realize there is an antenna, but what does the repeater look like, where is it, and how is it powered?
Do amateur repeaters sit in a rack at the towers?
Depends. Most repeaters themselves just look like a base station radio, some are rack mounted, some arn’t. The duplexers can be rack mounted as well, depending on the specific type.
Yes, most are housed in the buildings at the base of the towers. In some instances the repeater can be in a rack mounted cabinet mounted on the tower itself.
For instance, google search for the following
kenwood tkr750 or tkr850
Vertex vxr7000
Motorola mtr2000
GE MASTR II repeater
These are all repeaters, just packaged a bit differently.
Duplexers come in different flavors as well. Searching for “notch duplexer” and “bp/br duplexer” will show you what some look like.
Duplexers are magic, through filtering they allow the repeater to transmit and receive at the same time, using 1 antenna.
Power is normally supplied by a 12v power supply plugged into grid power if the repeater doesn’t have its own internal power supply.