I've seen it pointed out that the main point of functional programming is immutability, and that the benefits mostly flow from that. I haven't really learned much of any lisp dialect, but my (admittedly fuzzy) general perception is that this is also the preferred way to work in them, so my guess is that's where the benefit in reliability might come from.
Correct. If things are mutable, then in most languages, there can be spooky action at a distance, that mutates some field of some other object or does so indirectly via some calls. This then can change how the thing behaves in other circumstances. This style of programming quickly becomes hard to fully grasp and leads to humans making many mistakes. Avoiding mutation therefore avoids these kinds of faults and mistakes.