Why does Olivia agree to see the messenger in Act 1, Scene 5 of Shakespeare's Twelfth Night?
The main reasons why Olivia agrees to see the messenger Cesario, in Act 1, Scene 5, seems to be simply intrigue and curiosity. For one thing, Cesario certainly made quite a strong-willed impression on Olivia and Malvolio by absolutely refusing to leave. When Cesario is told Olivia is sick, he says he knows and insists he will speak with her. When Malvolio tells Cesario that Olivia is sleeping, he says he knows that too and will still speak with her. What Cesario is saying here is that he is aware of any excuse Olivia could possibly make not to speak with him or any other man, but upon Duke Orsino's instructions, he is absolutely determined to speak with her.
Another reason why Olivia finds the messenger intriguing is his description. He's described as being too young to really be a man, but too old to still be a boy, and part of what makes him too young to be a man is that he has no Adam's apple. In other words, he's not developed enough to be considered a man, as we see in Malvolio's lines:
Not yet old enough for a man, nor young enough for a boy; as a squash is before 'tis a peascod, or a coding, when 'tis almost apple. (I.v.147-149)
All of the images and metaphors Malvolio says in these lines are used to describe unripe fruit and liken Cesario to unripe fruit, showing us that he looks underdeveloped for a man. It is after Malvolio delivers these lines that Olivia gives permission to let him in, showing us that she is both intrigued by his persistence and his physical description.