Pure and chaste would, of course, refer to a love of one person by another that did not involve physical contact.
"From afar" means that the person only loved the other from a distance, possibly without the object of his or her affection being aware their admirer's existence.
At the very least, the object of affection might know the other person, but not the fact that they were loved.
This situation occurs often in literature, often with a man of lower class admiring a woman of high station, even royalty. Any romantic relationship between them would be strictly forgidden, so the man cannot even approach the woman for any reason other than normal interaction between upper class and a servant, say.
In this instance, though, the words mean somthing different.
The song is "The Impossible Dream" from "Man of La Mancha":
To right the unrightable wrong,
To love pure and chaste from afar,
To try when your arms are too weary,
To reach the unreachable star.
In the song, the singer believes they achieve a nobility or higher purpose by loving without the love being "dirtied" by any physical contact or urges.