Learn English – the meaning of “If you can’t sleep, don’t count sheep. Talk to the shepherd.”

The writer of the "missed connection" note is grasping at straws (<- useful English expression) to create some kind of connection to the person whom he encountered on the L train. He knows nothing about her other than that she was reading a book and had some bruises. "I read a book once" is a clumsy, funny way of saying, "Hey, I have something in common with you! You obviously read books, because I saw you reading. And you know what? Some time in my past, amazingly, I also read a book! (So that's enough for you to contact me and maybe we can go on a date.)"

In that sentence, the verb will is used with the sense “To choose to (do something), used to express intention but without any temporal connotations (+ bare infinitive)”. That is, while one might wish to be possessed (that is to say, under the influence of powerful emotions), typically one cannot be possessed just by deciding to be so.