I recently read Jane's fame by Claire Harman and the answer I seem to have gotten is that it happened slowly. It took time for people to appreciate her wit and stories about every day life. There aren't any daring escapes, adventures or passionate romances in her books yet, the simplicity makes them all the more tantalizing. In short, they are a breath of fresh air (in my personal opinion).
During her life time Jane Austen did not make a big splash and enjoyed modest sales - though it was not for lack of trying as Harman points out. After her death even her family did not think her novels would ever see the success they see now.
It's quite the accomplishment that over a century after her death Jane's books are still in publication, and have been frequently re-adapted into movies and other mediums. Not to mention the large fan-club devoted to all things Austen. Her stories have managed to transcend time and remain relevant in today's modern world. Yes, I do believe that the spin-off novels might be a bit silly and the people who solely enjoy the movies but have never read the books are missing out. However, at the end of the day the heart of her stories lives on and I can't begrudge people the way they choose to consume Austen's work.
Harman dispelled one of my own misconceptions that I had about Austen. I had been under the impression that Austen preferred to be an anonymous author and wrote as a hobby. When in fact Austen worked tirelessly to get her books published and took criticism very hard. Jane Austen's story also shows how hard it is to predict the longevity of a piece of work. Will popular works like Harry Potter remain a household name in 100 years from now? Or will they become a blip in literary history? It's hard to know what the future will bring (so keep calm and write on).
If you are interested in reading more about Jane Austen's life and rise to fame I would encourage you to start with Harman's book.