The server in this context is a "warmed" Function VM. The first time a VM instantiates, it goes through all of the setup required to run a function. That includes generic reusable stuff like Operating System setup, but also Function-specific things like JIT compilation of the Function software, library loading, language loading, all of which are specific to the Function being executed.
The Function VM is hyper-optomised to that one Function's task. There's compute-time costs in changing its role from Function A to Function B. In the public cloud, the only cost you save on is starting up the OS, if both Functions happen to use the same OS. It's faster in most public FaaS infrastructure settings to just destroy a Function VM than it is to transfer a VM from Function A's configuration into Function B's configuration. Similarly, it's faster to just leave the VM running for a little while, just in case the Function is invoked again.
When the paper discusses VMs idling, that's the scenario they describe, where servers (or more accurately software VMs inside of physical servers), are locked into a specific function for some number of minutes. These minutes are dead computing time.
Meta, as a private cloud provider to themselves, can do some interesting things to reduce the cold-start of their Functions, meaning they can run more efficient clouds if they choose, by massively reducing the idle-time that's an industry standard in the public cloud.