While it appears that the Talmud is prescribing rules of courtesy, this passage goes beyond the issue of propriety. Interrupting another person is not merely rude, but also unhealthy.
Cardiologists have described a "Type A" personality, which they find to be a significant cause of coronary heart disease. Among the characteristics of Type A people are the following: operating under pressure of time, doing multiple things at the same time (e.g. eating breakfast while talking on the phone and also reading the morning news), and finishing other people's sentences. The latter indicates not only impatience, which itself demonstrates the pressure under which they are operating, but also a presumptuousness, since they are taking for granted that they know what other people intend to say.
Teaching ourselves to allow other people to finish their sentences is a simple way to learn patience. Once we achieve it, it becomes easier to correct other Type A behaviors, such as making a mad dash to enter an elevator before the doors close completely, losing our composure in congested traffic, or feeling oppressed by the approach of a deadline. We may learn to take life in stride and even to relax, thereby eliminating the stress factor that has been implicated in heart disease.
No wonder that Solomon referred to the Torah as "a Torah of life." Adhering to its guidelines can actually prolong life.