Understand the rules to break the rules
As a rule about rule-breaking, you can break any rule of writing so long as you're doing it for effect. Broken rules suddenly work when the emotional effect evoked in the reader by breaking the rules is precisely the effect you want.
Which means that to understand how we might get away with mixing first- and third-person POVs, we'd better understand how each one "breaks" the other.
I wrote an article on what POV is for last year that covers this in some depth, so I won't rehash it all here. Suffice it to say that if you compare first- and third-person styling, you find this:
The difference between these POVs is fundamentally a difference about emotional distance and information control.
Understand those two axes, and the answer to that "when it works, why?" question becomes pretty straightforward.
Emotional distance
First-person narration has the viewpoint character telling you the story directly, in their own voice. Often, the character also lets you in on their inner-most thoughts. It is a very intimate narration style, and brings readers emotionally close to the viewpoint character.
By contrast, third-person holds them at distance. In third-person narration the reader's view of the characters is mediated by an (often anonymous and disembodied) narrator. The presence of that intermediary puts one level of separation between you and the characters, which in turn tends to diminish your emotional connection to them.
Information control
Third-person narration shares information pretty freely with readers. This is particularly so when the story uses an omniscient narrator, who is free to tell you anything, at any time, about anyone or any event in the story's world. Third-person narration has a way of laying all its cards on the table, as it were.
First-person narration, however, necessarily limits the information to only what the viewpoint character knows. Since the character is telling you the story, they can't tell you anything they themselves aren't aware of. All kinds of things can be happening--and typically are--outside of the first-person narrator's view. It should be pretty easy to see how a writer can use that limitation to create mysteries and otherwise hide stuff from the reader.
Guidelines for POV mixing
With that in mind, a story could benefit from using a mixed POV style if any of the following are true:
You want to pull one character very close, while leaving the other as an enigma. The obvious mode for that is to let readers be very close to a first-person hero, while keeping the third-person villain more distant. Because readers have access to the first-person hero's thoughts and motivations, they should have an easy time empathizing with them. Likewise, because we'd witness the villain and their dastardly deeds without the same understanding of why they're doing that stuff, readers should have no problem in hoping the villain eventually gets caught.
The story's major hook involves learning the antagonist's motives and/or plans. A first/third person split between protagonist and antagonist keeps the villain mysterious: because all we can see is what they're doing, but not at all why, we become curious to learn the reasons. That's the hook.
You want to deeply explore the villain's motivations and plans, while keeping mysterious the way in which the good guys are going to stop them. For this, you'd switch the viewpoints, putting the bad guy in first-person while keeping the good guys at distance. This would work when you have a villain who is a "true believer" in the rightness of their cause, even if their beliefs are totally screwed-up.
As far as how to mix them, I have two suggestions: continued here:
https://plottopunctuation.com/article-how-to-mix-first-and-third-person-pov.html
