Why Do People Say “Axe” or “Aks” Instead of “Ask”?

It's the Black English Vernacular (formerly Ebonics).

Correctness in language is a matter of custom, not logic. Not only does "aks" get the pass for common enough usage, it's also a shorter more distinct sound than the aspirant "ask". Paul Mooney said the black man is the most imitated man, and he's right. Every culture exhibits its own linguistic rhythms, and at the risk of being labeled a reverse racist, I find the cadence and trills of the Black English Vernacular the most hypnotic.

If you are interested, listen to some McWhorter lectures on linguistics—he will explain why this happens and how it remains part of a dialect or language. Many word spellings or pronunciations that are part of “proper” English started out as “poor” pronunciation or spelling 150-200 years ago.

For the same reason people say “anyways” instead of “anyway”, they grew up hearing it spoken that way.

When I was in college a few people pointed out my bad grammatical speaking habits. It requires focus to undo what one has spent years saying.

Poor education, and also peer-pressure from similarly badly-educated peers.

They weren't corrected in the first place when they mispronounced it wrongly.