I regret I am so late to the party, but I can't resist offering my two pence.

I completely agree that part of the answer lies in Mr Knightley's explanation, "his own mind full of intrigue, that he should suspect it in others," but I think there is more to it.

At the ball in Chapter 38, Mrs. Elton (speaking to Jane Fairfax) praises Frank within his hearing.

"At this moment Frank again began talking so vigorously, that Emma could not but imagine he had overheard his own praises, and did not want to hear more."

Mrs. Elton then tells her husband what she was telling "Jane," which Frank also overhears, and the following exchange with Emma ensues:

"Jane!" repeated Frank Churchill, with a look of surprise and displeasure. "That is easy; but Miss Fairfax does not disapprove it, I suppose."
"How do you like Mrs. Elton?" said Emma in a whisper.
"Not at all."
"You are ungrateful."

In this exchange Emma is undoubtedly teasing Frank about Mrs. Elton's flattering comments about him, but Frank's thoughts are all on Jane, not Mrs. Elton. In my opinion, Frank believes Emma is referring to Mrs. Elton's compliments and attentions to Jane (including sending the carriage). To Frank, "full of intrigue," Emma is hinting that she is aware of his secret.