What does Ethereum have to do with surfing a decentralized Web? Or like what is it bringing to the party? I would imagine most of this is just IPFS, right? Aren't there other like non cryptocurrency based distributed dns systems? I'm just confused how cryptocurrency popped up, or what am I just totally fucking missing?
You can't have decentralized DNS without some form of shared state, where the shared state must resolve the problem of "if I use key X to change the owner to Y, but then later on someone hacks my key and sends a message to change the owner to Z, how does the world know that Y came first?". This is equivalent to the double spending problem, and so a solution to this problem implies blockchains or some other architecture that can support a cryptocurrency.
A lot of the other responses here are too complicated and specific. Here's my attempt to put it into easier-to-digest terms:
In a traditional web stack you have a backend and a frontend. The frontend is the stuff the browser runs, and, simplifying a bit, the backend is everything else.
Ethereum smart contracts basically let you replace your backend logic and database with code that runs on the Ethereum blockchain network. Depending on your application can decide to run only a few parts of your backend on Ethereum, or the entire backend.
It's very slow when compared to traditional backends like nodejs, etc, but it has the benefits of censorship resistance and excellent availability. Better still, you don't need to run or maintain servers to support it if you don't want to (although there are benefits to doing so).
In this case, they're using Ethereum's replacement for DNS, ENS.
