Tap the "New Game" button, which slides away to reveal a tray of targets beneath. If the "New Game" button isn't visible, it means that a game is in progress. You can tap the "Finished" button in the lower left to clear it out and start over. Tap the "i" info button in the lower right to flip it over and set the options for your game.
Tap a ray bay (one of the 32 five-side red shapes around the outside) to fire a ray. In order to fire precisely the one you want, you can hold your finger down on the edge and an orange halo will appear around the ray that's going to fire when you release. You can slide your finger along the edge to position this halo, releasing on the ray bay you intended.
Rays travel in a straight line out of the bay, unless stopped, bounced, or returned by intervening targets. When the ray has finished its journey, the ray bay is covered with a marker indicating what happened:
This Wikipedia article has an excellent explanation with more detailed examples.
Lower scores are better: the object is to find the hidden targets with the least number of rays fired. Each marker that appears around the outside of the board adds one to the score. Thus, a ray stop and a ray return each add one, but a ray bounce requiring matching markers will add two.
When you are finished placing each of the targets in the tray where you believe they are, tap the "Finished" button in the lower left and red crosshairs will rotate in to indicate where the targets truly were hidden. Each missed target adds five points to your score.
Starting a game with the "show ray trails" handicap will double your score; seeing exactly where the rays travel is a significant assist.
Starting a game with the "keep targets off edge" handicap will also double your score. Using both handicaps will quadruple it.
You cannot change handicaps or the number of targets once a game is underway.