Why can’t prepositions be used as something to end sentences with?
Prepositions are fine to end sentences _with_. Prohibitions against preposition stranding are nothing anyone should listen to. (actually an infinitive marker) It is a made-up rule that does not reflect the true state of the language, so it is nothing anyone should worry about. Worrying about it is also something they (because English lacks a singular genderless personal pronoun) should not suffer from.
Look it up! (actually an adverb)
However, the misattribution to Churchill is something with which we should not put up. Even though he did not say it, it is still something no person should put up with.
Some sentences cannot be reworded to avoid stranding the preposition no matter how hard you try to.
The chair was sat on. (passive)
What is this tool for? (copula)
This is what he came back to me with. (fused relative pronoun)
This is something to look at. (infinitive phrase)
The first one simply has nothing it can be replaced with. The object of the preposition is the subject, omitted for the second time.
The second one cannot be reworded as *For what is this tool? because it must be For what purpose does this tool serve? As another example, This is what the tool is for becomes expanded to This is that which the tool is for. It cannot be *This is that for which the tool is, so it is due to the copula rather than the fused relative pronoun.
The third one is ungrammatical as *This is with what he came back to me unless it is a question such as With what did he come back to me?
The last one is awkward when corrected with ?This is something at which to look. because at which to look might mean that to look at.
The practice of preposition stranding is something that even the University of Oxford’s English Department approves of.
However, when doing **formal writing**, it is usually better to _front_ the prepositions instead of using them as something with which sentences are ended. Other than that, feel free to end sentences with prepositions without worrying about grammar complaints whenever you want to.