In the words of Shigeru Egami, master of karate, “The word itself (rei), however, can be interpreted in several ways; it is the rei of reigi, meaning "etiquette, courtesy, politeness," and it is also the rei of keirei, meaning "salutation" or "bow." The meaning of rei is sometimes explained in terms of kata or katachi ("formal exercises" and "form" or "shape"). It is of prime importance not only in karate but in all martial arts. For our purposes here, let us understand rei as the ceremonial bow in which courtesy and decorum are manifest. He who would follow the way of karate must be courteous, not only in training but in daily life. While humble and gentle, he should never be servile. His performance of the kata should reflect boldness and confidence. This seemingly paradoxical combination of boldness and gentleness leads ultimately to harmony. It is true, as Master Funakoshi used to say, that the spirit of karate would be lost without courtesy.”
Simply said, rei emphasizes courtesy in the martial arts, and without it, the spirit of arts such as karate would be lost.