It doesn't explain it. The only hint I see is here:

In Obscura, we only control the first hop and use blind third-party relays for the exit hop. The result is that we can never correlate your identity with your traffic as we don’t even see your traffic in the first place!

It sounds like they're using some kind of onion routing.

They also say they'll be open sourcing the client which should tell all about the privacy claims.

We’ll be open-sourcing all of our client code so you can look at the source code and verify that we’re doing what we say.