Here is the actual section on prepositions, and it's well worth reading the whole thing.
PREPOSITIONS have Government of Cases; and in English they always require the Objective Case after them: as, "with him; from her, to me."
The Preposition is often separated from the Relative which it governs and joined to the Verb at the end of the Sentence, or of some member of it: as, "Horace is an author, whom I am much delighted with." "The world is too well bred to shock authors with a truth, which generally their booksellers are the first that inform them of." This is an Idiom which our language is strongly inclined to; it prevails in common conversation, and suits very well with the familiar style in writing;
[Any objections so far, dear reader? Notice the "strongly inclined to"?]
but the placing of the Preposition before the Relative is more graceful as well as more perspicuous; and agrees much better with the solemn and elevated Style.
[That's it. Doesn't sound like a rule to me. It's a little stylistic tip for a particular kind of writing. And just to show that he understood, I shall quote further.]
...But in English the Preposition is more frequently placed after the Verb, and separate from it like an Adverb; in which Situation it is no less apt to affect the Sense of it, and to give it a new Meaning; and may still be considered as belonging to the Verb, and a part of it. As to cast is to throw; but to cast up, or to compute, an account, is quite a different thing: thus to fall on, to bear out, to give over &c. So that the Meaning of the Verb, and the Propriety of the Phrase, depend upon the Preposition subjoined.
Lowth's book has also been blamed for the (utterly fatuous) idea that you can't split an infinitive, which he doesn't mention once. In fact, he toddles along merrily beginning sentences with Ands and Buts and being a good clear writer of English.
So Lowth is less read than condemned. And next time anybody tells you that you can't end a sentence with a preposition, tell them to off piss.