That is a weird one, yes. I think it comes from 'flesh out', which, if similar, is at least old enough that it doesn't sound strange to me.
Wiktionary sort of suggests it should sound similarly odd - it doesn't discuss it on the flesh out page [0], but for flesh vb. [1] it does give '5. To put flesh on; to fatten'.
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That is a weird one, yes. I think it comes from 'flesh out', which, if similar, is at least old enough that it doesn't sound strange to me.
Wiktionary sort of suggests it should sound similarly odd - it doesn't discuss it on the flesh out page [0], but for flesh vb. [1] it does give '5. To put flesh on; to fatten'.
[0] - https://en.wiktionary.org/wiki/flesh_out#Verb
[1] - https://en.wiktionary.org/wiki/flesh#Verb