So, you want to strike when the iron's hot, and take advantage of that brand. You want to get Flappy Bird sequels coming out, you want Flappy Bird stuffed animals for children, Flappy Bird backpacks, etc. You want these things happening when the game is in its peak, and if you try to do everything on your own, you won't make that deadline, and Flappy Bird will be quickly forgotten. So, you get investors, bring in millions, hire staff, and start pumping out Flappy Bird everything to keep it in the spotlight and to grow the business. Or you start doing spinoffs, Wiggle Frog, etc. Either way, you need money to get things moving... fast.
Secondly, you need to fight off competition. This isn't a huge concern with Flappy Bird, but let's say you make a simple app that let's you post a few photos of yourself, and others vote on your best picture. You launch, it's a huge success, and you have practically zero expenses. Now, you're going to have a dozen competitors fire up overnight, and some of them will have more talented developers, some of them will have more free time, and some of them will have huge budgets. They want to take your crown, so you need to stay one step ahead of them. This is where investors come in. The money goes into polishing the app further, turning out new features, or related apps.