Historically, it was quite common for individuals to marry cousins. It was only after the sciences became more developed in the 19th century (1800s) that the risks of near-relation marriages were recognized. As proof, Charles Darwin himself was married to his first cousin Emma Wedgewood. One reason for the custom of marrying cousins was that wealth and estates would stay within the family if cousins married cousins.
One famous marriage between cousins was that of England's Queen Victoria to her first cousin Prince Albert. Albert Einstein was married to his first cousin Elsa Einstein. A very contemporary instance of love between cousins is the romance of Italian/English actress Greta Scacchi with her cousin Carlo Mantegazza. many Western countries still have no ban on cousin marriages, for instance, England has no law banning cousin marriage but is considering one.
In other words, cousins falling in love and marrying is only odd in terms of contemporary American culture. It was not previously odd and in many countries in the world today it still is not odd; it is in fact still quite common in many countries.
For one thing, something we have to understand is that society of the early 19th century didn't have the same reaction that we do to cousins getting married. If the cousins didn't grow up in or near the same household, they weren't seen as unmarriageably close relatives. Fanny and Edmund are indeed first cousins, but their first years of life were in such different backgrounds that they're not sibling-like cousins.
In the beginning of the novel, Sir Thomas and his wife (with Mrs. Norris) discuss this very thing: they decide that Fanny is actually less likely to fall in love with and marry Edmund if she is brought to live there. But when Fanny arrives at Mansfield Park, she is meek, terrified, lonely; Edmund is the only one who shows her compassion and kindness. Initially, her feelings toward him are not those of sexual or romantic attraction: she is just grateful that someone is nice to her. Eventually, this deepens and ripens into genuine concern for Edmund's well-being (in that Fanny thinks Mary Crawford is not good for him), which in turn becomes romantic love.
Austen Edmund's falling for Fanny in about two sentences, saying that she prefers to allow the reader to affix any timeframe onto this process. Perhaps Edmund and Fanny were somehow meant for each other: perhaps they could have met at any time and in any context, and they would have eventually ended up together. Or perhaps (and this is a reading that I favor), Edmund and Fanny are two kindred spirits who took a long time to grow romantic attraction to one another. Both of them seem to feel things slowly and carefully.
Try to ignore what would be the social implications of that today, since they were very different back then. Ultimately, Edmund saved Fanny from sadness and loneliness in her childhood, and Fanny saved Edmund from an imprudent marriage in his adulthood, and they owe each other fidelity and affection after all of that.