I'm looking forward for the comments about this. I discovered/read about "pet names" a while ago, but I could not understand very well what problem they're trying to solve, and if they solve it.
A naming system that is both permissionless and globally unique will tend towards unreadable names. You'll see lots of Sybil attacks on these systems, with botnets squatting any decent name.
For DNS, you have many governance bodies that act as gatekeepers for domain registration. Repository hosts like GitHub avoid Zooko's triangle by having namespaces, and an appeal process if someone is squatting your trademark. Crates.io on Rust doesn't allow automated crate creation.
In many cases, naming systems are not permissionless, and gatekept by a central entity. Once you open the floodgates of permissionless-ness, you're dealing with a hostile environment that's difficult to control.
Petnames are one solution to this issue. Others are Handshake, Namecoin and ENS
The second piece to this is how the GUI works. Instead of showing long bit strings or hex encoding them in some pointless way, it allows you to assign names to these addresses as you come across them. When you see the address in some other page it will show up with the name you assigned and so you will recognize it.
A key point to this is that HTTPS does not actually cause you to know you are using the right domain name. Aside from the fact that HTTPS is untrustworthy due to the CA model, mybrand.com or mybrand.net or one of the other infinite number of legitimate sounding things could all be the correct domain for some brand, you would never know unless someone previously introduced you to the right domain name. Since you had to get it from somewhere, you may as well have received the public key at that point, which could be encoded in a QR code, or, on the internet, which is the main use case of petnames, trivially included as a link in some web 8.0[1] page or chat where you first heard of the brand.
1. I guess it will take them 6 more tries to get this right.