The term has somewhat different connotations in modern times, but the idea behind it is fundamentally the same. It was the idea behind present day debutante or prom occasions.
In the 18th century (and even up to the last century in America) a young lady came out when she was of marriageable age - about 16- and officially presented into society for the first time (principally the gentlemen), usually with a ball in her honour if her family was of consequence. Young ladies of the aristocracy were presented 'at court' in St James' probably before the reigning monarch and broadcast in the leading newspapers; a priveledge Miss Anne de Bourgh in Pride & Prejudice is unable to partake in due to her "indifferent" state of health.
A woman was considered marriageable from 15 to 30 although if you were still an 'old maid' at 27 you were generally considered on the shelf.
In MP Fanny finally realises her claims to 'notice' and admiration when her cousins leave Mansfield and she gets a chance to step out of their shadow - at 18 which was rather late in life . The ball at Mansfield was for her growing cosequence but as she was not brought up in this "trade" she didn't even know that all eyes were especially upon her, scrutinizing her 'air', manner, countenance, beauty and of course dress and coiffure.
Also known as 'opening her career' or her 'first campaign' this practice often included a 'season' in London or Bath where the lady would be introduced into fashionable society, attending balls, concerts, operas, forming new acquaintance, picking up tips in the latest fashions, bagging a new wardrobe - and. hopefully a few beaux, possessed of good fortune.
There were a certain traditions associated with coming out. For example in some families younger sisters wouldn't come out before the elder were married, a custom which Elizabeth Bennet interestingly doesn't seem to agree with.
The married sister could then take tje place of the mother or another experienced female in chaperoning the younger ones to balls etc.
Another milestone in a young lady's life was when she reached the sweet age of 17 and was considered a grown woman and finished her education.