As simple as the game seems it utilizes so many of the concepts that addiction psychology is based on. The game is literally designed to get the mind hooked in the same way drugs affect the brain.
People love simple things. The simpler a product is, the more likely it is to become popular. Flappy bird is hard to master but extremely easy to learn. It has almost no learning curve and after a few seconds a person can become aware of all the game controls.
The combination of easy to learn but impossible to master creates a tremendous amount of frustration. One of the core emotions involved in making any game popular is frustration. The more frustrated a person gets the more he is challenged to try again.
Nostalgia is a powerful marketing tool and Flappy bird utilizes that concept perfectly. As soon as you see the graphics of the game you will unconsciously remember the beloved Super Mario and feel good without knowing why.
The game gives players the illusion of control. It makes them feel as if they have total control over the character while in fact there is a great lack of precision in controlling the bird. This illusion of control makes most people feel like they can beat the game the next time.
Just like slot machines get people hooked by making them believe they were about to win, Flappy Bird does the same. The bird usually dies a few millimeters away from the exit. This gives the player the feeling that he was about to win. See why do people gamble.
Humans are lazy by nature. The easier a game is to play the more likely it is to become popular. People can play flappy bird during times when their other hand is busy.
Games that end fast encourage people to play during the free short breaks that they take. If a person has 2 free minutes then he won't play a game with a long story line, but he will go for the fastest thing he can do during those minutes.
The human brain ignores repetitive patterns but gets interested in novel stuff. The Flappy bird character looks totally new. It's similar to a fish but it's not a fish. It looks like an evil character from the super Mario game but it's not. See why Super Mario got very popular.
Internet users and specifically smart phone users are much more impatient than the normal person. Fast paced mini games appeal a lot to this large segment of people.
A stubborn competitive person will hardly be able to resist getting addicted to Flappy bird. The game is designed to trigger the competitive spirit of a person, then use his stubbornness against him to keep him hooked.
The game colors give people pleasant feelings on the unconscious level. From a neuroscientific point of view, games with the correct color combinations make people feel great. Candy Crush is a perfect example.