I don't want to interrupt the loveliness of this idea, but there are bits from the video and the comments that I want to note:
Someone commented that this might make places under the rings where it was permanently dark. But because the earth is tilted on its axis and the sun appears to move north and south through the sky seasonally, so would the shadow move. But it's likely some places (the equator, generally everywhere between the tropics) would get some level of shade from the rings. The layer is very thin; aside from under the rings at equinox, there would be plenty of bright sun. Would there be enough insolation change to affect climate? Probably. But given the scenario, I expect the earth would have evolved this way, it's not like it would be a change from how things are now.
The person who made the animation seems to have designed it for equinox. Seasonal variations aren't addressed at all.
The theoreticals (e.g. one commenter's note that this would affect satellites) are irrelevant to me: It's beautiful.
I collect natural history, environmental, and geography books, almost compulsively. I look for used versions where I can (to save paper) but also appreciate that many of these books come out in runs of a few thousand copies, and aren't profit-makers for the publishing companies, so when they come out new, I buy them while they're new. If you like Geographile and want to let me know, please either comment on my posts, or visit my Powell's wishlist to feed my book collection. You might find books you like there, too.