In the game, players control a small fleet of ships, and because they're pirates — friendly cartoon pirates, mind you — they're going to sail the seas sinking ships and stealing loot. Players choose one of the scenarios in the box, and each scenario contains a number of quests, with each quest being comprised of one or more adventures. After you complete all of the adventures in a quest, you sail to Tortuga to spend your loot for ship upgrades, new sailors, or gold (also known as victory points).
Adventures are divided into easy, hard and crazy, and for each adventure you reveal a card from the appropriate deck to see what you're facing. In general, easy and hard adventures challenge you to take out various merchant and naval ships (which are represented by dice, just as your ships are), while crazy adventures are mini dice games that you play directly against opponents. For the easy and hard adventures, you see what you face, then decide how many of your ship dice to send against these targets. The active player drops all of the dice into a box, then players take turns moving or firing cannons before finally resolving battles in order based on which ships are closest to one another.
Thus, you have the luck of the die rolls affecting everything in two ways — which numbers or symbols land on top and where everything lands in the box — with you trying to mitigate that luck by using sails to move strong ships into better position or weak ships to safety, by using cannon to take out targets before a closer ship can beat you to it, and by grabbing sailors and specialized upgrades to let you do things that no one else can do.
For more on the gameplay in Rattle, Battle, Grab the Loot, and a view of the components — with the dice and artwork being final and everything else being prototype quality — here's an overview video of Chevee Dodd at Origins 2015: