Back before the Internet, BBS systems were accessed by dialup modem connections. The caller would dial the phone number of the BBS, where it would connect to a modem (and then to the computer through the modem serial port).
A simple BBS might have one phone line, so only one caller could connect at any given time. If the phone line was busy, you would try again later.
A larger system might have more than one phone line, and therefore more than one modem. Sometimes the BBS operator would install extra serial port cards in a single computer to control multiple modems, or other systems would have one modem per computer and the computers would talk to each other via a LAN.
If there was more than one phone line, a simple setup would have different phone numbers to call for each line. A more sophisticated setup would have a single "rollover" number set up with the phone company which would automatically find the next available line.
The software running on the BBS would communicate with the caller through the serial port. To draw screens, the BBS would send ANSI escape codes which pretty much all terminal programs (that's the software the caller runs) supported. The BBS could then move the cursor around, draw lines, use colours, etc.
A "door game" was a separate program from the main BBS. When the caller chose to enter a door game, the BBS would effectively unload itself, and run the door game program, telling it to communicate with the same serial port. The BBS would be out of the picture for a while, until the caller chose to exit the door game and return to the BBS.
With the advent of the Internet, all this changed because you no longer needed a point-to-point modem connection for each caller. Many callers could simultaneously connect over the same net connection. Of course, the BBS software had to change a bit to accommodate this new way of working, and the whole mechanism that door games were run had to change too.
There are a handful of traditional BBS systems still out there today, but I doubt any of them actually support dialup modem connections. They probably all support telnet connections, using appropriate software that actually understands the internet.
I'm not sure whether there are any particular resources I could point you to, but I happen to know a thing or two about doing this from writing BBS systems a few decades ago.