Fiend of Shadows has an ability that triggers when the Fiend deals combat damage to a player. When this ability resolves, the player that was dealt damage chooses and exiles a card from his or her hand. The controller of Fiend of Shadows can then play the exiled card.
The effect that allows the controller of the Fiend to play the exiled card does not specify that they can do so without paying its mana cost. This means that in order to cast the card, all costs must be paid as normal. For example, if a copy of Altar’s Reap were to be exiled, the controller of the Fiend must pay 1B and sacrifice a creature in order to cast the exiled card.
There are a few other things to be aware of with Fiend of Shadows’ ability. The first is that this ability only allows you to play the exiled card. For example, it is not in your hand, so you cannot discard it. The Fiend’s ability can exile more than one card. You must follow all timing rules associated with playing any of these exiled cards. You can only cast an exiled sorcery during a main phase of your turn while the stack is empty. Similarly, you can play a land that was exiled this way, but only during a main phase of your turn while the stack is empty and only if you have not already played a land that turn. Finally, the effect from Fiend of Shadows does not end once the Fiend leaves play. If its triggered ability exiles a card, you still have the option to play that exiled card even after the Fiend is no longer on the battlefield, because the ability that exiled it specifies that you can do so “as long as that card remains exiled.” It doesn’t care whether the source of the ability is still there.