For what I understand, a provider who trust a user, stores in her browser a set of kind of certificates, that basically declare "I "provider NNN", states that this user has been authenticated by me".

Another party will ask for one trusted token, verifies that it is really signed by provider NNN, and can trust that each time this token is resent, it is the same user.

The privacy is respected on the user side has the third party has no idea who is this user. The trusted tokens could even have a short life. Third parties therefore should have great trust in the trust token provider.

It is as if Google authorized other parties to use its own cookies.

Indeed this means that other parties trust "provider NNN". I guess that Google wants to become a more central identity provider than today, where it has to share this role with Facebook and a few other companies.