Coins can be used to buy Hints (revealing one word on a grid), to reveal the clues for Mystery Words, and to pay to complete Quests, uncover Trophies and reveal words in the Secret Message.
To buy a hint (reveal a word on the grid): simply tap the orange icon (showing four pips in a row) on the status bar at the top of the game screen. The price in coins to purchase a hint will be displayed.
To reveal the Mystery Word clues for the current grid section: tap the pink "?" (question mark) icon at the top of the game screen.