The meaning of "feel like old friends at first sight, and their eyes become a book" means: it is as if two people have known each other for a long time, and they seem to see the past and understand each other from their eyebrows.
Feel like old friends at first sight: Old friends meet and feel close to each other.
Browsing eyes make a book: The gaze and communication between two people are like a silent book.
In general, this sentence describes the tacit understanding, mutual understanding, and spiritual connection between two people.