I don't think they're referring to multiple threads when they say different part of the program, but instead that you should be able to reason about mutation locally.
For example, if I pass an argument into a function, it may be 'unexpected' that the argument is mutated - I can not reason about that mutation locally (unless it's very explicit or a known idiom such as push).
However, within a function, avoiding mutation seems pointless as you should have no trouble reasoning about it. At some point you really are just throwing away performance with significantly diminishing benefits.
Shared mutability across threads is definitely a huge pain in the ass though.
In the end I think we're all just trying to reduce the state space we have to manage in our heads when we read and write code, and removing mutability reduces that space.