This is not, incidentally, stunningly rich, although it is nonetheless rich. Mr. Darcy, for instance, makes 10,000 pounds every year; his friend Mr. Bingley makes about 5,000 pounds a year because he has inherited 100,000 pounds. Georgiana Darcy, Darcy's sister, has an inheritance of about 30,000 pounds, and Emma Woodhouse is in the same range. This would put them comfortably into millionaire territory today. While he's probably not the wealthiest person in the novels, the wealthiest character whose income we know is Mr. Rushworth from Mansfield Park, who is even wealthier than the very wealthy Mr. Darcy, making at least 12,000 pounds a year. Probably the richest character in the novels is Sir Thomas Bertram, although we're never told his income. If so, this would make Edmund's decision to be a clergyman that much more drastic: he's settling for an income of about 700 pounds a year (for comparison, Colonel Brandon in Sense and Sensibility is of the opinion that a bachelor living alone can scrape by on 300 pounds a year, so this is just enough to marry and keep house and not do much else); no wonder Mary Crawford, with her 20,000 pounds inheritance, is frustrated with him, and bad as it was, it's perhaps not wholly surprising that she has a secret wish for Edmund to be the eldest son. Henry Crawford, in the meantime, has an income of 4,000 pounds a year, which makes him a sort of Mr. Bingley, and probably about three Wentworths; given that Fanny Price can look forward to practically nothing, it's not surprising that people are astounded that she refuses him. In any case, Anne Elliot and Captain Wentworth probably could look forward to about 1,000 pounds a year in income, which is not a crazy amount of money, but enough that a responsible married couple could certainly live comfortably. James Heldman has more information about incomes in Austen novels here.
In any case, I was thinking more about how Captain Wentworth got the money. The money is prize money. The Napoleonic Wars were going on in this period, and in order to encourage the capture of French ships, it was customary to reward the crews for every capture. This allowed the government to encourage captures without any cost to itself -- the money from the capture would just be divided up. This handy Wikipedia chart shows the standard prize divisions (you can click it to get a better view):
The captain would typically get about 2/8 of the prize money, and the rest of the money would be divided up among the crew according to rank, as shown above. (Technically the Royal Navy still does prize money of a sort -- although there hasn't been much occasion for it -- but instead of dividing it up this way, it now goes into a common fund for all naval personnel.)
Wentworth received his position by taking a frigate -- a fairly state of the art ship -- with a sloop, which he commanded as a lieutenant -- which was largely just a patrol ship; while not unheard of, it was an astounding feat. He then becomes captain of a frigate, and does well enough with captures. The amount he comes home with, 25,000 pounds, is not a huge amount in terms of prize money, but we are told that Wentworth was very extravagant with his prize money in his early years -- this is actually one of the original obstacles Anne and Wentworth faced, because despite doing very well, Wentworth had managed to spend practically all of his prize money at a time when it looked like there would be little more coming, despite his confidence that he would get more. He's proven right, but this time around he learns to save and thus we have the 25,000 pounds. It's still probably only a small portion of his lifetime prize money, though.