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I often read the expression “would of” used instead of “would have”. Each time I read it I get annoyed so I googled it and found out -as I expected- that it is an incorrect way to say “would have”. Now, there are a lot of brilliant slabs words/expressions, so my question is, why do people use this one? It’s so annoying to read, stupid and clearly wrong, it has no clue, why did they came up with this expression?

Correction: what annoys you is people writing “would of” when they are saying /ˈwʊdəv/, which is the standard pronunciation of the contraction would’ve.

The vowel of the preposition “of” is almost always reduced in actual speech, yielding /əv/. Thus “would’ve” and “would of” are homophones. So no surprise that some people spell it that way, even though it makes no grammatical sense.

Would’ve can be even further reduced to /ˈwʊdə/, which some people spell woulda. The same goes for the modals, shoulda, coulda, musta.

Spelling as it sounds can yield amusing results:

Along the way the details of his past are sordid out and he realizes that what he once thought about his parents isn't the truth at all. — Amazon.com Review.

A speaker of British English, of course, would never write sorted in this manner, but with an American flapped t, it’s a perfect fit.

This is probably a case of hearing a phrase and assuming/guessing how it should be spelled. Would have can be abbreviated as would've, and in rapid conversation, the pronunciation of "would've" is basically the same as "would of."

"Would of" is a garden variety malapropism (Wikipedia - Malapropism).

Some more interesting malapropisms are "tantrum bicycle" instead of tandem bicycle, "Alcoholics Unanimous" instead of Alcoholics Anonymous, "a vast suppository of information" instead of repository of information, "Miss-Marple-ism" instead of malapropism¹ and Mike Tyson's "I might fade into Bolivian" instead of oblivion (these are all borrowed from that same Wikipedia article).

The basic idea is that no one has perfect knowledge of any language, not even the ones they speak natively. We hear things incorrectly and then repeat the mistake.

We know that English speakers often contract "would have" into "would've." This is pronounced identically (in some dialects) to "would of," so the mistake is easy to make.

1: This one seems too perfect to be a complete mistake. The "miss" sound is totally absent from "malapropism" and the term, for those who didn't follow the Wikipedia link, comes from a character named Miss Malaprop. It seems unlikely that the supposed speaker of "Miss-Marple-ism" wasn't aware, at least subconsciously, of the correct word, or at least its origins. In which case, this neologism may really be an eggcorn.