The coins appear automatically when you win a game. When earned you can you use the coins for power-ups. Those are the icons that appear at the bottom of the screen. The bomb eliminates two false answers, the timer gives you 15 extra seconds and the > will let you skip to another question and the other one will give you two chances to answer the question. You can only use 1 power-up per spin.
You can earn coins by winning games of Trivia Crack and reaching new levels. Sometimes you'll get an offer and you'll be able to use one power-up for free. Watch out for spam emails because sometimes you'll be offers which promise you a lot of free coins most of the times those are spam emails. Only focus on the coins you receive in the game.