Well a router takes a packet from one of its interfaces and uses the Internet Protocol address encoded in the packet header to determine which of its interfaces to forward the packet to based on the preconfigured destination subnet for the interface. Most home routers only have two interfaces and two subnetworks.
Most people are familiar with home routers which do Network Address Translation and have a built in Wireless access point. Neither of those things are required for something to be a router. In fact routers with IPv6 support do not perform NAT between your local network and the Internet for IPv6.
Similarly, DHCP, DNS, and various other things that home routers do can actually be handled by totally separate hosts on the network. That's what he's doing.
If you're looking for more information about how networking works, I would highly recommend Computer Networks by Andrew Tanenbaum. It's more abstract than the typical "Understand TCP/IP in 600 pages" books that are available but it provides a good high-level overview of how networking works, what protocols matter, and how everything fits together.
Edit: When he says he's using VMs, he means that he's using Virtual Machines to run multiple operating systems on the same server. Each of these operating systems runs one or more servers for DHCP or DNS or other networking services. I assume that he's using his virtualization platform to mirror the virtual machines between his servers and provide hot spares, so that if one VM goes down another spare can step in.