When you help someone by offering positive words of support and approval, that's encouragement. It must have been the words of encouragement you gave in the airport lounge that helped your friend board the plane.
Within the word encouragement we see the word "courage," which means the ability to face danger and deal with it. To encourage, then, is to help develop that ability in someone, while the "ment" at the end makes that development into a noun, the act of giving courage or support to another. You needed a little encouragement to try horse riding again, after your injuries healed.