No, the statement isn't nonsense. If variables always had the same value and "never changed" in Haskell, they wouldn't be variables. They could be hard-coded literals.

You're just mistaking a claim about variables that change each time a program is run (which means they are initialized to a different value each time, whether mutable or immutable) vs. changing during the execution of a single run, which would require mutability.

The quote was referring to the former, not claiming that mutability is required.