Sensibility is best demonstrated in the main character, Elinor, whose defining character trait throughout the book is sensibility, logic, sound reasoning, and a reigning in of passionate or extreme sentimentality.
Let's look at some of the times when Elinor demonstrated sensibility. She has a very pragmatic view of love; when she is separated from Edward, she realizes that it is probably because she has no money, and it isn't a prudent match. It stinks, but she realizes the truth of it, and tries to cling to that as she copes. When they move residences to Barton, she is very sensible about their finances and the reality of their situation. and, not citing specific instances, but rather her character as a whole, she is the voice of reason and moderation in comparison to her more volatile, passionate and dramatic sister. She constantly tempers Marianne's extreme nature.
Sensibility is also a defining theme of the novel. It is the sensible sister who, in the end, gets the man that she loves. It is also Marianne's sensible choice of Colonel Brandon (instead of the intense, love-doomed Willoughby) that in the end earns her happiness and a peaceful life. Many of Austen's female leads are the ones with sensible heads on their shoulders, and they are the heroins. She seems to feel that being rational and sensible was an overlooked trait, and one worth praising. I hope that those thoughts helped; good luck!