The characters combine to mean “to know books (知书) and thoroughly understand principles/etiquette (达理).” The structure implies a cause-and-effect relationship: because one has studied and knows books, one understands the proper way to reason and behave.
知书达理 is a cornerstone of traditional Chinese values, deeply rooted in Confucian philosophy. For Confucius, the ultimate goal of education (读书) was not merely to acquire facts but to cultivate one's character and become a morally upright person (君子, jūnzǐ). This idiom perfectly captures that ideal. It suggests that knowledge without morality and social grace is incomplete. A helpful comparison for Western learners is to contrast it with the term “intellectual.” In the West, an “intellectual” is someone who is highly educated and engages in critical thinking, but this doesn't necessarily imply they are polite, reasonable, or socially graceful. In fact, the “arrogant intellectual” is a common stereotype. 知书达理, however, has no such negative connotation. It is an unequivocal compliment that marries intelligence with social and emotional wisdom. It is the outcome of an education that successfully builds both mind and character.
This is a formal and very positive term. It's not something you'd use casually in everyday slang.