Although it's not a pretty sight, birds fly into airplane engines from time to time. These events are known as "Bird Strikes," and as many as 160,894 instances have occurred from 1990-2015 in the United States alone.
For the most part, Bird Strikes aren't dangerous for the passengers onboard an aircraft. Only 0.25% of occurrences lead to accidents, as pilots can still fly most commercial airplanes with just one operating engine.
From the inside of the airplane, a Bird Strike can be a scary event. You might feel turbulence or see smoke coming from the plane's engine. Shrapnel inside gets disintegrated from all the moving parts, causing a lot of the ruckus you'll see from inside the window.
Bird strikes are most common during the day between July to October and when the airplane is taking off or landing. They can happen under other circumstances, but it's uncommon.