Angry Birds had no intellectual property or previously successes to look at, but it was a breakaway hit. Here is a list of some of the things I think helped it become a real star.
1. An extremely simple primary interaction mechanic. Angry Birds didn't try to do anything crazy with the input device. Press, drag, release. It's important to note that games like Doodle Jump, Cut the Rope, and Tiny Wings all had a habit of keeping it simple.
2. A motivating, but simple, level structure. The first level in games like these should always show the simplest case that proves out the game. Angry Birds first level is "fling the basic red bird at a pig". It's simple, trivial, but still cute and easy. Cut the Rope also starts with easy first levels with a single, easy to understand "tutorial" instruction.
This is important.
It requires the essence of the game to be distilled to a single action verb like "cut the rope" or "fling the bird". Don't mistake this for a lack of potential complexity. Levels become more complex as you linearly move through time.
3. Increasing complexity. Once you unlock the "yellow bird", the "blue bird" and the "bomb bird", you gain a form of player package complexity (namely complexity you use directly). By this time you'll also be seeing different block and pig types to interact with.
4. OCD Bonus. This is a term borrowed from World of Goo, which took it very literally. The idea is that a given level should have a "win more" condition. This can be seen in World of Goo, Angry Birds, Peggle, Cut the Rope and many more game.
5. Gimme Advantage. My step-dad who works with cows and has never really played games once took my mom's iPad to beat a level she was struggling with. Some great games like Tony Hawk never really get here, since mastering the game at an input level is impossible without holding the controller for hours at a time. Angry birds is easy to understand by watching the person playing.