As has been mentioned on this forum earlier, the events of Emma are narrated almost entirely from the point of view of the eponymous heroine. On first read, we see events from Emma's perspective, and we interpret events in the same way she does. It's only on re-readings that we can appreciate the hints Austen drops throughout that Emma's judgments of events and characters are not to be trusted. This passage furnishes another example of the shifting realities of the situation and their presentation through Emma's partial eyes.
The facts are these: Jane Fairfax, an orphan, comes from a poor family. Her mother is Miss Bates's sister. But Jane has been raised by Colonel and Mrs Campbell as a companion to their similarly aged daughter. Jane has therefore received an excellent education, but she has no way of supporting herself. The plan all along has been for her to become a governess once she is 21 years old. Meanwhile, the Campbells' daughter has married and has moved to Ireland with her husband, one Mr Dixon. Jane was entrusted as a chaperone during Mr Dixon's courtship of Miss Campbell, so he and Jane are well acquainted. On a boat ride during a trip to Weymouth with the Campbells, Jane had been nearly thrown overboard, but Mr Dixon was able to grab her in the nick of time. Coincidentally, Frank Churchill has been in Weymouth at the same time, and has met the Campbells, Mr Dixon, and Jane.
After the Dixons marry and move to Ireland, Jane needs to start looking for a governess position. However, she has been ill, and the Campbells, who are genuinely fond of her, want her to recover before she takes a job. Meanwhile, the Dixons invite the Campbells and Jane to Ireland. The Campbells are eager to go, but Jane demurs. She says she will go back to her own family, i.e., to Mrs and Miss Bates, for three months, to recover her health and to spend time with them before she will be forced to move away for a governess role.
Emma dislikes and is jealous of Jane. As Mr Knightley points out, Jane has all the fine qualities Emma attributes to herself: "it was because she saw in her the really accomplished young woman, which she wanted to be thought herself". She is therefore always willing to believe the worst of Jane. Against this background, Emma learns that Jane Fairfax is going to visit her aunt Miss Bates and her grandmother Mrs Bates in Highbury.
The passage in question deals with about five different interpretations of this situation. First, what Jane Fairfax says; second, what the Campbells say; third, what the Campbells actually think; fourth, what Emma thinks; and fifth, what the actual reality of Jane's situation is. Let's look at them in turn.
- What Jane Fairfax says. Jane says that since she needs to start a job soon, she should spend her remaining free time in Highbury with the Bateses rather than go to Ireland. Also, since she is ill, her "native air" might help her recover. Finally, the Bateses can look after Jane in Highbury, whereas there would be no-one to nurse her in Ireland.
- What the Campbells say. The Campbells agree with Jane about the need to recover her health. They also agree with her that she should spend time with her family while she still can. So they reluctantly agree not to take her with them to Ireland.
- What the Campbells actually think. The Campbells see that it will be very difficult for Jane to suddenly have to work hard for her living. They wonder whether they have been unthinkingly cruel by accustoming Jane to a comfortable life, when they knew all along that it was unsustainable. They feel that if Jane must lead a life of hardship, it would be as well to let her start sooner rather than later, as there is no point in putting off the inevitable. There is also a hint that the Campbells, fond though they are of Jane, want to distance themselves from her because of their own guilt about the situation. They have raised Jane as their daughter, but cannot leave her money to secure her financial independence, because their money will be bequeathed to their actual daughter. These are the double and triple motives that the Campbells have.
- What Emma thinks. Emma is intelligent enough to figure out that there is more to the story than simply "Jane is ill, and needs to come home to get well." But as usual, she leaps to the wrong conclusion because of her own dislike of Jane: "an ingenious and animating suspicion entering Emma’s brain with regard to Jane Fairfax, this charming Mr. Dixon, and the not going to Ireland." That is to say, Emma assumes, with very little foundation, that Jane must have tried to break up the relationship between Mr Dixon and his fiancée, the Campbell's daughter, because she wanted to marry the wealthy man herself. She further assumes that the Campbells are aware of this failed seduction, and therefore do not want to take Jane to Ireland. Hence Emma's imputing double and treble motives to both the Campbells and Jane.
- What Jane Fairfax is really thinking and doing. The above answers the question as asked: what are the multiple motives at play with the Campbells. It also explains how Emma too ascribes multiple motives to the Campbells, but is mistaken. The remainder of this answer goes into Jane's motives. This is the sort of thing that becomes evident only on rereading, so it is not possible to explain without major spoilers. Please read no farther if you wish to avoid spoilers.
At one point, Emma thinks how annoying it is that Jane Fairfax, rather than Frank Churchill, is coming to Highbury:
Highbury, instead of welcoming that perfect novelty which had been so long promised it—Mr. Frank Churchill—must put up for the present with Jane Fairfax.
Like Jane, Frank has been raised by a wealthy family as their son. Unlike Jane, however, Frank stands to inherit considerable wealth from his adoptive parents, and has taken on their surname of Churchill rather than his birth name of Weston. Frank's stepmother, Mrs Weston, had been Emma's governess. Mrs Weston and her husband (Frank's biological father) wish very much for a match between Emma and Frank. For this reason, Frank is a subject of great interest, particularly to Emma.
Mr Weston has only recently remarried, and Frank has not actually met his stepmother. Frank keeps promising to visit, but invariably fails to turn up. Emma is eager for Frank to visit his father and stepmother so that she herself can meet him. Instead of this much-desired visit, Emma is forced to put up with Jane. On actually seeing Jane, Emma realizes that she has been extremely unjust. Given Jane's character, it is not possible that she would have set out to deliberately lure Mr Dixon away from Miss Campbell. But instead of repenting for this thought, Emma merely recasts it: Jane and Mr Dixon must have sincerely fallen in love, but Jane would not follow through, because that would be a betrayal both of her friend Miss Campbell and of the kindness that the Campbells have shown her. Having made up her mind (again without evidence) that this must be the case, Emma interrogates Jane on the subject. Jane refuses to engage with Emma:
She was more reserved on the subject of Weymouth and the Dixons than any thing. She seemed bent on giving no real insight into Mr. Dixon’s character, or her own value for his company, or opinion of the suitableness of the match. It was all general approbation and smoothness; nothing delineated or distinguished. It did her no service however. Her caution was thrown away. Emma saw its artifice, and returned to her first surmises. There probably was something more to conceal than her own preference; Mr. Dixon, perhaps, had been very near changing one friend for the other, or been fixed only to Miss Campbell, for the sake of the future twelve thousand pounds.
Emma not only persists in believing that Mr Dixon and Jane must have been in love and in badgering Jane for evidence. Annoyed by Jane's refusal to furnish any such evidence, she concludes that the reason for his having married Miss Campbell after all must have been her money, rather than a noble sacrifice on Jane's part. Having thus confirmed her dubious conclusion to her own satisfaction, Emma promptly puts aside all thoughts of Jane's interesting situation and presses her for information on the one subject that looms large in her own mind: Frank Churchill. Jane again refuses to reveal any information:
The like reserve prevailed on other topics. She and Mr. Frank Churchill had been at Weymouth at the same time. It was known that they were a little acquainted; but not a syllable of real information could Emma procure as to what he truly was. “Was he handsome?”—“She believed he was reckoned a very fine young man.” “Was he agreeable?”—“He was generally thought so.” “Did he appear a sensible young man; a young man of information?”—“At a watering-place, or in a common London acquaintance, it was difficult to decide on such points. Manners were all that could be safely judged of, under a much longer knowledge than they had yet had of Mr. Churchill. She believed every body found his manners pleasing.” Emma could not forgive her.
Emma pushes the conclusion that Jane and Mr Dixon are in love with each other. But the reality, glaringly obvious on a second reading, is: Emma is barking up the wrong tree.1 Jane has come to Highbury because she and Frank Churchill, having met at Weymouth, are secretly engaged to each other. Both Jane and Frank have ostensible reason to be in Highbury, because their families of origin are there. Jane refuses to go to Ireland because by coming to Highbury, she can meet Frank when he makes his long-deferred visit to his father and stepmother. Jane has no intention of working as a governess. Her health suffers under the strain of her secret engagement, because she is too upright a person to carry off such a major deception without feeling pain. She deflects Emma's questions about Weymouth because going into any detail would endanger the secrecy of her relationship to Frank Churchill, which developed there.
Like everybody else, Jane has multiple motives here: the stated ones, and the very dangerous secret one. Only Mr Knightley sees through the smokescreen. He even warns Emma. When Frank finally comes to Highbury, Mr Knightley tells Emma that it's noteworthy that Frank has visited Highbury only after Jane Fairfax is ensconced there. The blind, brash, and selfish Emma, sure that Frank is more drawn to her charms than to Jane's, laughs off Mr Knightley's hint. She and the reader soon realize Mr Knightley is correct. Austen has set a trap, and both Emma and the reader have fallen into it.
1 I'd revise to remove the image of Emma's pushing a conclusion while barking up a tree, but now I find it amusing.