Jtdcftul

"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 Mrs 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.

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."

What most of you are missing is that "of" is preposition, and prepositions are slippery beasts. The "rules" for prepositions are complex and, for most people, in large part incomprehensible.

While an English purist would quickly cry "foul!", someone with, say, a 5th-grade education might easily believe (without applying much critical thought) that, in "If I had the time I would of eaten sooner", "of eaten sooner" is a prepositional phrase which somehow modifies "I would".

They are speaking/writing without applying an English teacher's "starch", and to them it makes perfect sense. After all, that's how (they think) their parents speak.

Great question. I also get annoyed when I see this phrase, especially from people who should know better. At one of my old jobs year ago, I worked with an account manager who actually used that phrase in an e-mail, saying something like "I should of known better". Yes, an account manager "should of" had a better education to know proper grammar... or at least know that's not a phrase used by professionals.

Even though the phrase might be pronounced and heard as "should of" (or "would of" or "could of"), there's no such phrase in written English. No school that I've heard of teaches this phrase and basic rules of grammar "abbreviate the word have as 've".
And writing "should of" takes just as many characters to write as "should've", so it's not like it's text-speak.

In my opinion, it's people being stupid, ignorant, or trying to be funny/ hip/ trendy by using the wrong phrase. Then other people see it and want to be in on the joke, so they use it also.

It's simply because, in many dialects, the sounds of "'ve" and "of" in "I would've bought two of them" are very similar or identical. People often confuse words that sound the same: there/their/they're, your/you're, etc.

In addition to KarlG's answer, there's a paper that argues that speakers in some dialects have actually reanalyzed the reduced have as "of" acting as a complementizer, i.e. working similarly to the 'to' in English infinitives, giving the bracketing I would (of worked). The gist of the argument is that have cannot further reduce from [əv] to [ə] while of can always reduce from [əv] to [ə]. However in the construct under discussion, the supposed reduced version of have can in fact reduce from [əv] to [ə]. Therefore, according to the paper's argument, the [əv] in [aɪ wʊd əv dən ɪt] is not actually have, but of.