"[...] up with which I will not put."

Solution 1:

• This is the sort of English I will not put up with.

"With" is a preposition (and it's "wrong" to end a sentence with one.)

• This is the sort of English with which I will not put up.

"Up" is still a preposition, so we gotta change it again.

• This is the sort of English up with which I will not put.

And there we have Churchill's response.

(PS. I am ignoring whether it's actually his response, or his altered response, or what have you, and focusing on the initial question. :) )

Solution 2:

It sounded stupid because the normal way of saying that is "this is the sort of English I won't put up with" which ends in a preposition.

He was making a point that sometimes it is better to end a sentence with a preposition rather than slavishly following alleged rules. (Alleged because it IS ok to end a sentence with a preposition.)

Solution 3:

As I learned the quote, he didn't have a problem with the "English" involved, but with the prescriptivist busybody who had red-pencilled one of his speeches.

The quote, as I learned it, was "This is the sort of arrant pedantry up with which I will not put."