Looks like these are being setup in libraries/stores/etc where cell service is bad, so I dont think hes using a cell modem.
I'm betting hes connecting to the local buildings wifi and routing calls that way via VOIP, voice over IP. So these look like a simultaneous donation, his payphone/tech/effort and the locations power/internet.
@Sempf @chartier he may use an Arduino to control the power/signalling to the payphones and other linked equipment.
He appears to be using as much of the original tech as possible - payphones need various extra signalling on top of standard analogue lines - and maybe some other type of ATA (analogue telephone adapter) for the VOIP end (a commercial off the shelf one would be easiest way of dealing with things like battery and ringing voltage, and I believe some of the Grandstream ones even handle loop disconnect (pulse/rotary) dialling).