Private DNS is what Android calls DNS over TLS. It's basically normal DNS but with a TLS connection wrapped around it.
DoT is very easy to self host if you already run something like a pihole (using nginx to proxy a tcpstream + having it wrap a TLS connection around it) and can be exposed to the internet because it can work over TCP (thus reducing the DDoS risk factor significantly).
In Android there's a setting to enable it in the network settings. The default will be "off", if you pick "on" you'll probably be using Google's DNS servers, if you pick "hostname" you can pick a different server.
It's more than that, private DNS is not just a different DNS server, it's a DNS over TLS (DoT) server. This means encrypting the lookups to prevent the ISP from tracking the host names you visit.
Many DNS servers don't support DoT and some support DoH (DNS over HTTPS) instead.
they recommend leaving it on because then all your dns queries go to google and no one else by default--their "private dns" defaults to the very unprivate google dns servers.