We find out offhandedly a bit more about Colonel Brandon's family in situation, through dialogue as is Austen's wont. We find out for example through Mrs. Jennings' solicitude that he has a sister, that he has about 2000 pounds per year (which isn't highly highly rich, but is quite well-off, as best I can figure out), and that he inherited his estate from a brother, rather than a father, making him a younger son who inherited unexpectedly. (And we even get a hint about what this older brother may be like, considering Brandon inherited the estate in a bad shape financially.)
And of course the Dramatic Moment of Willoughby's leavetaking. Obviously not all is well in Denmark, or England in this case, but it remains for future chapters to find out just what's going on, or if it all works out. He partakes of a bit more silliness before he leaves though, with his declarations of the perfections of a smoky, smaller-than-ideal cottage. (Although I can't think it too bloody small when it has two garrets!)
We also see Elinor basically stifling logical fears because of her mother's romanticism, or at least that's one way to look at it. All I can say is that Elinor's father must have been a man of good sense, and she took after him, because otherwise it's hard to figure out where she got the more sedate temper and logical nature when surrounded by her mother, Marianne, and Margaret! Also: an excess of romantic delicacy is not a good parental trait!
We do however get shown Marianne's inner good nature, buried as it is sometimes beneath the silliness - she's got to be genuinely upset to have her hopes raised when she thinks Willoughby has come back to her, only to have her hopes dashed, and instead of being peevish and sulky, she's genuinely glad to see the person once she realizes it's her sister's love, despite the fact that she doesn't think much of him in a Romantic sense. And if I haven't made it clear yet, I think this is one of Austen's strengths as an author - in short, you're supposed to show, not tell, and Austen does that quite well. IMHO.
"If we find they correspond, every fear of mine will be removed.": At the time, it was very improper/Not Done for an unmarried girl/young woman to write letters to (and receive them from) someone to whom she wasn't engaged (or related by blood or some such.) So cousins and brothers/sisters might correspond, and engaged couples, but not just any two young men and women.