Usually the attacker, on their own computer, or some other server they have root on, will open a port and expose it to the internet and listen. The exploit payload will then make an outbound connection to that port. Once it's connected, the exploit will give the attacker's computer shell access. Search terms include 'reverse shell'.

It takes the normal client/server architecture and turns it inside out. If you remember FTP and active vs passive, it works like active mode FTP.

That's just one way to do it. If the attacker wants to actually listen on an open port on a compromised server that's behind a firewall, look up 'NAT traversal' for like half a dozen ways to do it.

One interesting method to get a shell that I read about is (ab)using ICMP echo requests. ICMP echo requests can contain arbitrary bytes as a payload. So the exploit will poll the attacker's IP address with ICMP echo requests. The exploit will have data payloads that have the shell's output. The attacker's server will respond with ICMP echo requests that have whatever the attacker wants to type into the shell. It's kinda janky but it works. Lots of firewalls might block outbound UDP/TCP connections from internal servers that don't need to make outbound connections, or might whitelist the addresses they're allowed to connect to. But they won't block ICMP, either because it's considered harmless or they forgot or they didn't know it needs to be blocked separately with other rules.

The point is there's any number of ways to do it, each more clever than the last.