Reasons of the importance of Mansfield Park at its time or in its society? What is the relevance that makes it such a important and key book?
It is actually difficult to argue the points of importance and relevance because the reception to this novel in Austen's time was cool. Reviews were brief and neutral without great criticism yet equally without great praise: it was responded to as an insignificant novel. There were two exceptions, yet these exceptions did lead the trend in criticism throughout the nineteenth century.
The chief novelist of the period, Walter Scott, gave Austen ringing reviews claiming her literary kinship with Shakespeare and other earlier great English writers. Later, George Henry Lewes gave a similar review praising Austen's writing and talent and comparing her to Fielding. Subsequent reviews and criticism then continued to follow the tone set by Scott and Lewes.
Austen always had a favorable readership and a popular following but is not considered to have been one of the bestsellers. The rank of bestsellers went to Victorian-style authors like the Bronte sisters, Eliot, and Dickens. These writers evinced the desirable qualities of Romanticism and Victorianism, those being the emotion and sentimentality and colorful descriptions that Austen disliked and avoided. In fact, Scott praised Austen's realism, marking Austen as one of the forerunners of the Realism movement. It was only in the twentieth century that Austen began to shine with the critical and popular lustre she is noted for today.
Some points that might help you argue the points of relevance and importance relate to her style. We've already noted her realism. In this novel, she employed an omniscient third person narrator in order to expose the thoughts and inner character of each of the principals, like Mary Crawford and Mrs. Norris. Austen reverses her previous trend and attempts an unobtrusive narrator; her effort isn't a complete one as part of Austen's style is ironic comment that illuminates character and plot development.
In addition, principal characters are all wealthy suffering one form or another of moral indolence and moral apathy. Exceptions are Mrs. Norris and Mrs. Price, respectively, who are of modest means and actual poverty. A very significant point of style and plot is that the heroine has been so overcome by circumstances and Mrs. Norris (!) that she is a heroine few like. Most find her weak as an individual and insignificant. Readers don't fancy Fanny Price the way they fancy Elinor, Marianne and Elizabeth.
Fanny Price was at this time just ten years old, and though there might not be much in her first appearance to captivate, there was, at least, nothing to disgust her relations. She was small of her age, with no glow of complexion, nor any other striking beauty; exceedingly timid and shy, and shrinking from notice; but her air, though awkward, was not vulgar, her voice was sweet, and when she spoke her countenance was pretty.