The four characters combine to mean “a family/gate (门第) filled with the fragrance of books (书香).” It's a highly elegant and evocative way to describe a family lineage built on scholarly honor.
The concept of a 书香门第 is deeply tied to China's history, particularly the Imperial Examination system (科举制度, kējǔ zhìdù) which lasted for over 1,300 years. For centuries, the primary path to social mobility, wealth, and power was through academic excellence. Passing these rigorous exams could elevate a commoner to the highest ranks of the government bureaucracy. Consequently, families that consistently produced scholars (文人, wénrén) who became officials were held in the highest esteem. This created a cultural ideal where academic achievement was seen as the ultimate honor for a family. A 书香门diethyl_ was not just respected; it was a model for society.
Even though the imperial exams are long gone, the term 书香门第 is still used frequently today with a very positive and respectful connotation.