It's a myth originally perpetrated by stuffy Victorian-era English teachers, and propagated by stuffy English teachers ever since. The root of the "rule" is actually from Latin - in Latin, you really can't end a sentence with a preposition. Somewhere along the line, some pedantic putz decided it would be a good idea if such a rule existed in English as well. The trouble is, as the citation in your post notes, that often has the effect of making sentences sound really stupid when they purposefully avoid prepositions at the end. The best rule of thumb is that if one can easily avoid prepositions at the end of sentences in formal or scholarly writing, do so. If they can't be avoided easily, or it's writing or speaking in any context other than the formal or scholarly, don't worry about it.