Dear Miss Austen, don't look now, but your backstory is showing
Austen opens this chapter with two separate digressions:
1. A word about the Bennet's fortune (or relative lack thereof) and
2. A word about Mrs Bennet's relatives.
The Bennet family income
Hey, remember the opening of Sense & Sensibility, where Austen goes on at length about the estate being entailed and the females being unable to inherit? Well, the situation with Longbourn is just the same, except instead of the estate going to a worthless half-brother, it is set to go to an estranged distant cousin that none of them has met. While Mr Bennet is alive, his wife and daughters have a home; once he dies, they are all out on the street, and would have to scrape by on Mrs Bennet's income (interest from her 4,000 pounds) and possibly live with other relatives or become governesses. Unless, of course, the girls marry, in which case they will have the support and protection of their husbands.
Mrs Bennet's family
Turns out that Mrs Bennet is not the daughter of a gentleman. She is the daughter of a tradesman - an attorney in Meryton. Just to get this out there right away (and hopefully to help you keep people straight later), her maiden name was Gardiner. Mrs Bennet has a brother - Mr Gardiner - who lives in London with his wife; he runs a booming business. She also has a sister, Mrs Philips, who lives in Meryton with her husband, who took over the law practice once owned by Mrs Bennet's and Mrs Philips's father.
The regiment is coming! I do so love a redcoat!
The regiment that is encamped at Meryton is part of the militia, essentially the national guard, and not part of the professional army. Although people will persist in saying that Austen never wrote about politics and the news of her times, this would be her doing just that. The regiments were deployed about England for training and because for much of Austen's life, they were under threat of invasion by France. First Impressions was first written in about 1796, when England was still dealing with the French Revolutionary wars. By the time it was published in 1812, England was caught up in the Napoleonic wars with France. There was a brief armistice 1803, but it didn't last long.
Having a regiment visiting town brought a good deal of menfolk into the neighborhood - many of them young, unmarried gentlemen acting as officers. It was considered a perfectly acceptable thing for gentlemen who either had little fortune or who were younger brothers and therefore not in line to inherit the bulk of an estate to serve in the militia. It's little wonder the girls are all a-titter - so many lovely strange men in uniform!
Mrs Bennet's Machiavellian designs
Jane has received a last-minute invitation to join Miss Bingley and Mrs Hurst for dinner, since the gentlemen are going into town to dine with the regiment. She (sensibly) would like the use of the carriage, so she can go and come back again, but Mrs Bennet is having none of it: "No, my dear, you had better go on horseback, because it seems likely to rain; and then you must stay all night."
That's right, she is deliberately sending Jane over to Netherfield Park in such a way that it will be impossible for her to come home at night. Way to impose on the neighbors, Mrs Bennet. Also, way to contrive for Jane to be there until morning, when she might actually see Mr Bingley.
Mrs Bennet considers it a personal triumph when it does indeed rain. The rest of her family still thinks she's off her nut, more or less - Lizzie and Mr Bennet even more so when, the following morning, they get a note from Jane saying that she's taken horribly ill. As one does when one gets wet. Look, it was a perfectly acceptable explanation for things at the time and a well-used (almost hackneyed) literary device, so just roll with it.
Jane is, in fact, sick enough that her hosts are insisting that the apothecary come to see her. Actual doctors were (a) thin on the ground and (b) expensive, and recourse to the local apothecary was quite common in Austen's day.
Let's go to the play-by-play, shall we:
Mr Bennet: Well, my dear, if Jane dies from this illness, it will be a comfort to you to know that it was all under your orders and in pursuit of Mr Bingley.
Mrs Bennet: (repressively) "People do not die of little trifling colds." This is awesome news, by the way, since now she'll have to stay there until she's better. I AM A MATCHMAKING GENIUS!
Lizzie: O_o. Right. I'm walking to Netherfield to check on Jane.
Mrs Bennet: You'll get muddy
Lizzie: I'll be fit enough to see Jane, and that's really all I care about. Three miles there, three miles back. I'll be home before you can say "Bob's your uncle." Or in a trice. Whichever is shorter.
Mary: I am saying something pedantic that would have been better conveyed as "You are overly exuberant. Get a grip."
Lizzie: *walks to Meryton with Kitty and Lydia, then it's hop, skip and a jump (or twenty - but she is literally hopping, skipping and jumping) to cross a bunch of pastures and get to Netherfield.*
[If you aren't certain what a stile is, you can check out this prior post, which includes a video of people crossing one.]
Lizzie: Good morning all! I'm here to see Jane. How is she?
Miss Bingley and Mrs Hurst: Miss Elizabeth, we are . . . surprised to see you. Also, as you can see we are in the middle of breakfast. *think evil thoughts about mud, early morning exercise, turning up too early for a morning call and the impropriety of such a long walk without a chaperone*
Mr Bingley: Hello! It is so lovely of you to come and enquire after your sister! Won't you make yourself at home?
Mr Darcy: *remains silent, thinking things like "her bright eyes are even brighter after all that exercise, and I'm feeling an unpleasant tightness . . . in my trousers" along with "Isn't it something like three miles to Longbourn? I'm not so sure a young lady should travel so far unescorted"*
Mr Hurst: grunts and eats his breakfast. I seriously wonder why on earth the elder Bingley sister married him in the first place.
And then Elizabeth spends the day with Jane, and the Bingley sisters are actually kind-hearted and solicitous toward her, and then when the time comes for Elizabeth to leave, Miss Bingley offers the use of her carriage (which is exactly the polite thing to have done), only Jane would rather have Lizzie stay so Miss Bingley is essentially forced into asking a second Bennet sister to stay on as a houseguest.
If you want to see some of this chapter play out, watch the following video to the 4:50 mark. (It does include dialogue not from the text, that sets the stage for what is yet to come - it takes a much harsher line on Jane and Elizabeth's social status than does the actual text, so you know. It also reassigns all kindness to Charles Bingley and removes any and all kindness from his sisters, because the filmmakers preferred to deal in absolutes, I believe, rather than the complex characters that Austen actually created.)
Tomorrow: Chapter Eight
Back to Chapter Six