We are told up front that Mr Knightley's dislike of Frank is "for some reason best known to himself", and Austen does not (yet) tell us what it is, but that it is somehow related to Emma is quite clear from the remainder of Mr Knightley's thoughts and comments in this chapter.
By happenstance, Mr Knightley (walking with Emma and Harriet) bumps into the Westons (walking with Frank Churchill) and Miss Bates (walking with her niece) - the latter two parties having met up already by chance - or is it? But I digress. Frank Churchill asks a question about Mr Perry, the local apothecary, getting a carriage, claiming that Mrs Weston mentioned it in one of her letters. When Mrs Weston denies any such knowledge or occurrence, Frank laughs and calls it a dream . . . except that Miss Bates knows it to be true, as does Jane Fairfax, who now has her head down, fussing with her shawl, while trying to avoid catching Frank's eye.
Once again, games pop up in Emma
Once inside Hartfield for tea, Frank Churchill seizes on a box of "alphabets" - hand-written scraps with letters on them used to form words - rather like doing a word scramble while using Scrabble tiles (indeed, it's a fine use of Scrabble tiles - you pull out the letters for the word, then set the lot of them in front of someone else, who is to solve the puzzle). Frank's first word goes to Jane Fairfax, and is revealed to be "blunder". Mr Knightley is then certain that Frank is playing, in Austen's term, "a deeper game."
Jane is embarrassed when Frank creates the word "Dixon", showing it first to Emma and then to Jane, and she sweeps aside without reading another offering from Frank - but she does not refuse his assistance in helping her to find her shawl.
Misunderstanding between Emma and Mr Knightley
Mr Knightley asks about the word Frank showed to her, and she is so embarrassed that she doesn't want to talk about it - it's a reminder of her suspicions regarding Jane Fairfax having a fling with Mr Dixon, and she doesn't want Mr Knightley to know she thinks it possible. Mr Knightley, however, believes that she is flustered because her affections are attached to Frank Churchill.
Yet he would speak. He owed it to her, to risk any thing that might be involved in an unwelcome interference, rather than her welfare; to encounter any thing, rather than the remembrance of neglect in such a cause.Poor Mr Knightley. Poor, hamstrung Mr Knightley, who believes that Emma and Frank are a couple. He will be laboring under this belief for many more chapters now, mistaken though we readers know it to be. And yet, the plot thickens very much upon us indeed.
"My dear Emma," said he at last, with earnest kindness, "do you think you perfectly understand the degree of acquaintance between the gentleman and lady we have been speaking of?"
"Between Mr Frank Churchill and Miss Fairfax? Oh! yes, perfectly.--Why do you make a doubt of it?"
"Have you never at any time had reason to think that he admired her, or that she admired him?"
"Never, never!" she cried with a most open eagerness--"Never, for the twentieth part of a moment, did such an idea occur to me. And how could it possibly come into your head?"
"I have lately imagined that I saw symptoms of attachment between them--certain expressive looks, which I did not believe meant to be public."
"Oh! you amuse me excessively. I am delighted to find that you can vouchsafe to let your imagination wander--but it will not do--very sorry to check you in your first essay--but indeed it will not do. There is no admiration between them, I do assure you; and the appearances which have caught you, have arisen from some peculiar circumstances--feelings rather of a totally different nature--it is impossible exactly to explain:--there is a good deal of nonsense in it--but the part which is capable of being communicated, which is sense, is, that they are as far from any attachment or admiration for one another, as any two beings in the world can be. That is, I presume it to be so on her side, and I can answer for its being so on his. I will answer for the gentleman's indifference."
She spoke with a confidence which staggered, with a satisfaction which silenced, Mr Knightley.