No need ta be so formal. Yer makin’ my neck itchy. Talk normal, got it? (Source)
I have no idea what does "Yer makin’ my neck itchy." mean. I guess the phrase has some figurative meaning.
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Possible reason: Until about 100 years ago, well-dressed men wore starched collars, at least in formal situations. These were very stiff and scratchy. Even now, when I'm wearing a tie with my ordinary modern business shirt, my neck gets itchy. (I don't usually wear a tie.)
I think it simply means, "You're making me feel uncomfortable."
It's not an established idiom, though; I'd guess it's a creative expression coined by the author. I could only find two instances of it on the Internet (the original story, and this ELL question). Moreover, when I tried to use a more formal form ("you're making my neck itchy"), Google said:
No results found for "you're making my neck itchy".
He's just saying it very informally, but what he's trying to convey is that he feels uncomfortable because the person whom he is speaking to is speaking formally.