1 Cute character
People get emotionally attached to certain characters fast. The cute little animal featured in cut the rope was one of the reasons people got attached to the game.
2 The character intensifies the connection with its actions
When the character is not fed it makes a sad face. When it eats it makes a happy face. These kind of actions make the emotional connection to the character even stronger and more powerful (See the importance of rewards in game design).
3 Challenging puzzles
While the game features an interesting character and bright colors it's not a simple game by any means. The game has very tough puzzles and sometimes people struggle for sometime before they can beat a level.
4 The use of physics
Cut the rope has so many elements that follow the rules of physics. Ropes swing if they carry a heavy item, balloons move up but can be pushed to the sides using air nozzles and many other elements that depend on physics that make the game very logical and interesting.
5 Bright cartoon like colors
The game has bright colors similar to the ones in cartoons. Studies have shown that those colors trigger pleasant emotions and a good mood. Even adults can experience mood improvements by looking at the screen, simply because those colors target the subconscious mind directly.
6 The use of candy
The use of candy and sweets throughout the game makes the game so attractive to the subconscious mind. Because humans like sweets and candies they get a pleasant feeling when they see them in a game (See why Candy Crush got so popular).
7 Level variation
Each few levels a new object is introduced that has different obstacles and tools. The fact that the objects keep changing keeps the game interesting. Since people get bored of repetitive things the novelty of the game makes it more interesting.
8 Feeding a little animal makes the game interesting
Many puzzle games look very serious and can sometimes be boring. The fact that the game revolves around feeding a small animal makes this puzzle much more interesting.