You've already described the meaning quite accurately - think of a situation where someone's facing a difficult situation or task, and other people doubt whether he or she will handle it. To express his/her (self-)confidence that the result will be positive, one could say
„Das schaffe ich schon!“ / „Das schaffst Du schon!“ / „Das schaffen wir schon!“
"Schon" may convey different meanings. The german "Duden" enumerates the most important ones. In your example, it would be #4 - expressing confidence, often in reply to doubts expressed by someone else.
As translation I would propose:
"I can handle it" / "I can manage it"
I think using "can" is stronger than "will" - as an attempt to catch the meaning of the "schon".