Why does its have no apostrophe but it's does?

Because it's is a contraction of it is or it has, and the apostrophe marks the missing letter — while its is a possessive pronoun, and possessive pronouns never take apostrophes.

It's raining = it is raining. The dog wagged its tail = the tail belonging to it.

The test is mechanical and never fails: read the sentence with it is substituted. If it still makes sense, you want it's. If it does not, you want its.

The reason this trips up even careful writers is that it appears to contradict the ordinary rule for possession, where an apostrophe is exactly what you add — the dog's tail, Sarah's book. But that rule applies to nouns. Pronouns form their possessives differently, and none of them use an apostrophe: his, hers, ours, yours, theirs, whose, its. Nobody writes her's, and its follows the same pattern.

Historically the two forms did compete, and it's was used as a possessive well into the eighteenth century. The split we use now settled later, which is why it can feel arbitrary — it is a convention, not a logical necessity.

The same confusion produces the whose and who's pair, and it works identically: who's is who is, whose is possessive.

Spellcheckers are of limited help here, because both spellings are real words. Reading a draft aloud with it is expanded catches almost every instance.

One further point: its' is never a word in any context. If you have written it, the intended word was its or it's.

The wider version of this error is adding an apostrophe to a plural — apple's £2 on a market sign. Plurals take no apostrophe. Apostrophes mark either omission or possession by a noun, and nothing else.