This looks like one of those wireguard based solution like tailscale or netbird though I'm not sure they are using it here. They all use a public relay used for NAT penetration as well as client discovery and in some instance, when NAT pen fails, traffic relay. From the usage, this seems to be the case here as well:

Share the local Minecraft server:

$ holesail --live 25565 --connector "holesailMCServer420"

On other computer(s):

$ holesail "holesailMCServer420"

So this would register a "holesailMCServer420" on their relay server. The clients could then join this network just by knowing its name and the relay will help then reach the host of the Minecraft server. I'm just extrapolating from the above commands though. They could be using DHT for client discovery. But I expect they'd need some form of relay for NAT pen at the very least.

As for exposing your local network securely, wireguard based solution allow you to change the routing table of the peers as well as the DNS server used to be able to assign domain name to IPs only reachable from within another local network. In this instance, it works very much like a VPN except that the connection to the VPN gateway is done through a P2P protocol rather than trough a service directly exposed to the internet.

Though in the instance of holesail, I have heavy doubts about "securely" as no authentication seems required to join a network: you just need to know its name. And there is no indication that choosing a fully random name is enough.