When that page loads, scroll to the bottom of the screen and look for the “Advanced” section. There you’ll see options for restoring different types of data.
After you choose the type of info you need to recover, you’ll be taken to a screen where you can pick what date to restore, say, or which file.
Pay close attention to the caveats at the bottom of the different restore screens! When you recover calendars, for example, you’ll have to reconfigure your sharing settings, any invitations will be re-sent (yuck!), and the recovered version will replace your existing info on all of your iCloud devices.
Here are the warnings listed on the “Restore Contacts” window:
For iCloud Drive files, you’ll be able to see how long you have to recover each document in question so that you can take action if necessary:
So, yeah. Just be sure you read the fine print before you proceed. This is incredibly helpful, though! I mean, everyone should be using an on-site backup of some type (you are, aren’t you?), so recovering contacts and documents should never have been a huge deal. However, with iCloud calendars, this’ll be a much simpler way to restore lost stuff, as Time Machine won’t allow you to just get things back up and running in the simple way you can with, say, contacts or email. So kudos to Apple.
If I were a snarkier person, I might comment that this has been a long long LONG time coming, but folks, I’m just not that petty.
(Note: If you have multiple Apple IDs and aren’t sure which one to use to sign in, check your devices. On the Mac, you’ll do that at System Preferences> iCloud; on iOS, that info is under Settings> iCloud. Whichever email address is listed in those places is the one you’ll want to log in with at iCloud.com, as that’s the one that’s syncing between your devices!)