The answer to this question can be found towards the beginning of the book as Winterbourne first gets to know the intriguing character of Daisy Miller. From where they are, they can clearly see the "far-gleaming walls of the Chateau of Chillon," which were made famous by the poet Lord Byron, and Daisy Miller expresses her desire to see it to Winterbourne:
"I want to go there dreadfully. Of course I mean to go there. I wouldn't go away from here without having seen that old castle."
As Daisy explains, they had planned to go the previous week, but her mother's illness prevented them. This of course gives Winterbourne the precise opportunity he has been looking for to get to know Daisy Miller better by offering to take her himself and give her a special guided tour.