I feel like a lot of answers here are dancing around why people find it offensive without really addressing it.

As an adjective “female” is completely fine to distinguish between genders when applied to humans. As in “a female athlete” or when a form asks you to select “male” or “female” (ideally with additional options “diverse” and “prefer not to answer”).

Where it’s problematic is when it’s used as a noun. In English “a male” and “a female” is almost exclusively reserved for animals. For humans we have “a man” and “a woman”. Calling a person “a female” is often considered offensive because it carries the implication of women being either animals, property or at least so extremely different from the speaker that they don’t consider them equal. This impression is reinforced by the fact that the trend of calling women “females” is popular with self-proclaimed “nice guys” who blame women for not wanting to date them when in reality it’s their own behavior (for example calling women “females”) that drives potential partners away.

So in itself, the word “female” is just as valid as “male” and in some contexts definitely the right word to use but the way it has been used gives it a certain negative connotation.

Female as an adjective is perfectly fine.

A female patient, a female politician, a female customer, etc. That’s the best way to refer to those.

What’s bad is using ‘female’ as a noun: "A female. "

In general, you just don’t use adjectives-as-nouns to refer to people. You don’t call someone “a gay”, “a black”, or “a Chinese”. That is offensive, and “a female” has the same kind of feel.

(there are exceptions to the above: you can call someone ‘an American’ or 'A German", but not “A French”. I don’t understand why - if you can’t feel your way, best just avoid it)

Now, you could get around it by calling someone “a female person” - except that we already have a word for “female person”, and that’s “woman”. And to go out of your way to avoid saying “woman” makes you sound like some kind of incel weirdo, and you don’t want that.