This is done through Ethereum smart contracts.
Here's some documentation on all this: https://docs.ens.domains/contract-api-reference/publicresolv...
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.
[edit: not that you have to use the crypto stuff with IPFS, but the same folks creating it also have had some sort of crypto currency use case]
IIRC they raised an embarasingly large amount of money during their ICO.