Why do some people say "aks" instead of "ask"?
Because of a regular sound process called metathesis — the swapping of adjacent sounds — and in this particular case the "wrong" form is actually the older one.
The history is the interesting part. Old English had both āscian and ācsian, and both were in use for centuries. Chaucer used axe. The form ax appears in early English Bible translations, including Wycliffe's. For a long period, ax was the dominant form in many regions and was entirely standard.
Standardisation eventually settled on ask, largely through the influence of particular dialects on the emerging written standard — not because one was more logical or more original. Aks is a direct descendant of the older variant, preserved in varieties that did not follow that particular standardisation.
Metathesis is a completely ordinary process, and English is full of words where it won:
Bird was Old English bridd. Horse was hros. Third relates to thrie. Wasp appears as wæps. Dirt was drit.
Nobody considers these errors, because the metathesised form became standard. The only difference with aks is which form the standard happened to adopt.
It also operates live in modern speech — comfortable to "comftable", nuclear to "nucular", prescription to "perscription".
Why it happens. The consonant cluster /sk/ at the end of a syllable is somewhat awkward, and /ks/ is easier to articulate in that position. Sound changes generally follow ease of production.
Why it attracts such strong reaction. Aks is strongly associated with African American English and with some British regional and Caribbean varieties, and criticism of it is consistently directed at those speakers rather than at metathesis generally. Nobody is corrected for saying "comftable".
This makes it a clear case of a linguistic judgement functioning as a social one — and a good illustration of how "bad grammar" claims frequently track who is speaking rather than what is said.
It is systematic, not careless.