"Why's that then? I didn't see any assignment of copyright to Slater, just an order saying the animal couldn't sue to claim that it held the copyright."
If, as Slater originally claimed, the monkey randomly wandered up and snapped a bunch if pictures including the famous the selfie, with this ruling there simply is no copyright. If that is true, when Slater made the picture public, it became "public domain" and anyone can freely use it. Of course Slater now claims he intentionally set up the shot, which confuses things a bit. If Slater wants to keep control of the picture, he probably has more legal fights ahead to see if a court recognizes his copyright claim.
BTW, contrary to another commentard's claim, the general rule in the US is that the person who pushes the camera shutter button does have copyright (look it up). If a photographer sets up a shot but has someone else push the camera button, at best that photographer can claim joint copyright with the button-pusher. The rule he is conflating is "work for hire", which usually overrides that intellectual property ownership rule and generally gives employers ownership of things like copyrights and inventions created by employees & contractors. But to be safe most serious professional photographers are careful to also have contracts that clearly spell out who gets the copyright - just like most R&D tech companies redundantly have developers sign specific, 'your employer owns your inventions' contracts.
Here is a slightly entertaining take on the subject (from 2014): https://www.theatlantic.com/politics/archive/2014/03/paging-bradley-coopers-lawyers-you-might-own-ellens-famous-oscar-selfie/358758/