There is not really an in-depth answer possible for this - this is also something that's still true today rather than simply a historical artifact. If a woman had a courtesy title (either Lady Firstname Lastname or the Honourable Miss Firstname Lastname), she would hold onto it when marrying a man with no title of his own, becoming Lady Firstname Marriedlastname or the Honourable Mrs. Marriedlastname. He would not be inherently elevated to the aristocracy, although this marriage might have been part of a social rise that would eventually lead to being granted a title from the crown.

You may be interested in this past answer of mine on the use of courtesy titles, and this one about commoners marrying aristocratic women.