DNS servers deal with domain names, not URLs. For an HTTPS request, the rest of the URL (path etc.) is only ever sent to the target site, over an encrypted connection. From an information privacy perspective, DNS is a weak link in this mechanism.
The reason running your own DNS improves your privacy has to do with how centralized or decentralized your DNS traffic is, and who gets to see it as a result.
If you use a DNS controlled by someone else, the DNS operator can see, and therefore (in principle) track, all of the domain name lookups you make that aren't served by your local machine or router's cache.
In contrast, if you run your own DNS server that's not just a slave to e.g. an ISP's server, then external domain lookups are made directly against the authoritative DNS servers for the domain you're looking up. There's no longer a single central server, other than your own, that's seeing all your lookups.
So if you look up news.ycombinator.com, and the name isn't already in your cache, the request goes direct from your DNS server to an AWS DNS server, because that's who hosts the ycombinator.com DNS. Your non-cached DNS requests are thus spread out across all the DNS servers for all the different domains you're accessing, and the only DNS servers that see your requests are the servers for the domain you're accessing.
There are some caveats to this.
First, if your ISP is unscrupulous, it can examine your DNS traffic anyway, since most DNS traffic is currently unencrypted (this is changing, but slowly.) To get around that you'd need an encrypted DNS proxy hosted outside your provider's network.
Second, large DNS providers like Verisign, GoDaddy, AWS etc. will tend to get a large subset of your requests anyway. However, most of those big companies are publicly focused on information security, and it would be big news if it turned out they were tracking your DNS lookups. Some of them provide audited guarantees that they don't do this.