I claim no substantial expertise here, but the way I think it works is:
Enabling DHCP on the router causes it to assign IP addresses on the local network (subnet), only, i.e. on its 'downstream' side. The addresses typically take the form of 192.168.0.xxx or something similar, using one of a few address ranges that are not valid on the wider internet, and are specifically reserved for use on local subnets.
For those local computers to access the Internet, the router also has to implement "IP translation", where the router modifies the address in packets going to/from the internet. To do that for more than one computer at a time, the router has to have more than one >internet visible< IP address, which it typically gets from your ISP by acting as a DHCP client or something similar, on its upstream side.
You may have to pay extra to get more than one IP address; somewhere in the fine print, the ISP should tell you the number of IP addresses that they make available. My previous employer had DSL with 16 IP addresses provided. They had more than 16 computers on the local net, and if you came in late, you could access the local server and peers, but you might not be able to get on the internet. You didn't get a warning message or any specific indication, you just couldn't connect to anything outside until someone else shut down their computer.
Mike Halloran
Pembroke Pines, FL, USA