How does Sam Know?

[Galadriel55]

Originally Posted by LadyBrooke

While we know that everything turns out all right, there was no reason for Boromir to think that two little hobbits could find their way into Moria and destroy the ring by thereselves.

Do you mean Mordor?
Also, in the book, when the Fellowship members hold their own counsil just before leaving Lothlorien, Boromir thinks aloud and says things that suggest that he wants to use the Ring as Gondor's weapon against Sauron. Although Sam was sleepy at the time, he could have heard and remembered the words, like he overheard Frodo's and Gildor's conversation in the Shire.