Why the "give" in “I don't give a flying f***”?

I’m not a native speaker. I know that I don't give a flying fuck means "I don’t care", but how did it come to mean that? Specifically, why does the verb give mean "don’t care" here?

Your question is a misunderstanding of give. Give doesn't mean don't care. The don't is already in the phrase:

I don't give a flying fuck.

Give: to bestow, especially officially; confer.

I don't give means:

I don't bestow/confer/hand over/offer/impart, etc.

What don't you give? Well, if you don't want to give anything of value, then you don't give a damn.

In the middle ages, there was a common saying not worth a curse. To damn someone is to curse someone to damnation. Not give a damn is to see something as unworthy of giving a curse or care over.

If you want to say it less emphatically, you can say

I don't give a hoot.

If you want to be more emphatic, you can say

I don't give a flying fuck.

Since a flying fuck is an imaginary thing to most people, this means you care so little that you wouldn't even give the thing something non-existent.

Basically, it means I don't care.

_{damn (v.) late 13c., "to condemn," from Old French damner "damn, condemn; convict, blame; injure," derivative of Latin damnare "to adjudge guilty; to doom; to condemn, blame, reject... the optative expletive use likely is as old. - Etymonline}