You know, I think if you ask different people from different fields (RF, analog, digital, PE, EMS's/PCBA manufactures) you'll get different answers. People working in higher frequency may quote EMI/EMC issues, and others may quote manufacturing related issues.
Like with any facet of engineering, there is probably a scenario that exists where the advise they give is only useful to them because of the environment or design conditions they find themselves in. It's not a particularly helpful answer, but, unfortunately it's true.
BUT, no matter what field you're in and if you are making PCBs, they have to be manufacturable. So, a couple of reasons PCB shops want copper pulled away from the edge are:
Corrosion - having copper come to the edge exposes it to oxygen, which can then kick start oxidation.. you can imagine the problems if all your plane layers are oxidising..
Short Circuit - kind of a specific sub point to the above, but, oxidation can cause tracking via dendrite formation, which then could cause a short on the board edge between layers. For instance, I attached an image showing this not on the board edge, but, on an external layer between pads.
Safety - Not a PCB fab reason but, having copper all the way to the edge also leaves the chance for something/someone to interact with the circuit, increasing chance of electrical shock.
There isn't one answer here where you'll leave this conversation going "okay, that provided the entire picture and now I know". It's more of a collection of answers given the field and environment you work within. But, like I said, the board needs to be manufacturable so hopefully the first two manufacturing related answers gave you some points to start looking into.
Here are a couple of links you can look into that will help kick start your investigation: