Based on their mention of "coax" I bet they have a Linux box w/ a modem doing dial-up PPP to an ISP, and a 10Base-2 NIC that they used to attach another PC. The Linux box was doing IP masquerading (NAT) to share the PPP connection w/ the machine(s) on the 10Base-2 LAN.
Having the IBM AT a a serial terminal would let somebody run CLI-based software on the Linux box (like Lynx, an IRC client, FTP, etc). You'd just be using a shell account on the Linux box.
I did stuff like this in the early 90s at home and later at a company I worked at (sharing a single dial-up connection over 10Base-2 with 5-ish Windows 95 PCs).