You asked how the crews would have used them. I never worked on the DRGW, but phone booths like that were common on all railroads in the days of trainorder operation and before radios were common. Here are a few ideas of how they would have been used.
Basically they provided communication between train, engine, maintenance of way crews, and anyone else out on the line and the dispatcher in the days before radio.
In the case of maintenance of way employees, they would call the dispatcher and get a "line up" of expected trains for some period of time, and use that information for planning their on-rail motorcar movements. If the maintenance of way crews were going to do work that would block the line, they would get authority from the dispatcher and inform the dispatcher when the work was complete.
Train crews would generally use the phone only if something was not going to plan. Generally they got trainorders from the open offices at Chama, Antonito (and Cumbres while it was still open). If everything went according to plan they would have little reason to use the phones along the line. But it was probably not unusual for something to happen that would cause then not to be able to fullfil their orders.....derailment, engine failure, etc. So first off they would call the dispatcher and inform him of the problem. Then the dispatcher could issue new trainorders directly to the train or engine crew.
The dispatcher was the nerve center for communications. If there was a derailment or other problem, he would get that information by those phones and notify who ever on the railroad needed to know. Or call in medical help, etc. Trainmasters, roadmasters and mechanical supervisors all had to make sure that the dispatcher knew how to get ahold of them in case of problems.
If there was a derailment out on the line you might visualize a line of supervisors waiting to use that thin line of communication to keep in touch with the dispatchers office. Those phone booths could become critical operations centers during emergencies.
Those little phone booths were probably the venue for a lot of interesting stories over the life the DRGW over Cumbres.
