I haven't read the story, but the meaning seems to be that the speaker feels emasculated because his wife supports him financially. The phrase "kept woman" dates back to an era when it was rare for a woman to be financially independent. When a man provided a house and an income for a woman not his wife, she was his "keep". The term "kept man" is a role reversal: in this case, the man is being financially supported by a woman.
The narrator feels humiliated at having to ask his wife for money. Perhaps in the early days of their relationship, the couple could laugh at the unusual reversal of roles by making a joke about the husband's being a "kept man". Part of the joke is that "kept" is an unusual word to apply to a man; another part is that one would not apply the term "kept woman" to one's wife. Over time, such jokes between the narrator and his wife have worn thin, and by this point, the husband can no longer laugh at such jokes.
(A WW II era injunction to secrecy made the same sort of joke, applying "kept woman" in an unusual context: "Be like dad and keep mum". "Keep mum" means "be quiet", but the pun relies on seeing "mum" as a kept woman supported by dad.)
"Vasectomizing" is a sanitized way of saying "castrating". A vasectomy is not castration, of course, but the narrator is complaining that he feels impotent. Even though he is (a) male and (b) a father, he has to ask his wife for money, which makes him feel that his manhood or virility has been taken away.
So the male narrator is just moaning about feeling humiliated because his wife is wealthy and he has to rely on her money to get by. He feels like a kept man, as he is being supported rather than being a provider; but he can't even provide the sort of sexual services one would expect from a kept man, because he feels castrated or at least vasectomized by his financial dependence on his wife. Quite an unattractive character, this whiny unreconstructed dude.