Once set up, when you try to log in to Google on any other computer or device, you just have to input your email address and ask to be let in. A notification shows up on your phone asking if you're the one who is trying to sign in from another device. Approve and ta-da, you're in. You can still use your password, of course, but it won't be required anymore if you're using your phone.
Google's email to Rohit goes into further details about the feature and some Q&A regarding it. You can check it out below and see if your concerns are answered in it.
This would clearly benefit those who often rotate their passwords or don't want to bother remembering them. It makes things easier for you, but shouldn't affect your security or privacy. If your phone is locked with a fingerprint reader or another secure method, there's little chance of this being abused by someone who accidentally or purposefully ends up with the phone in their hand.
In addition, Google confirmed that this feature is being tested for both Android and iOS users and that testers can still opt to enter their typed password in any instance where this feature is available.