Birds sometimes don't have the smarts to evade strong winds. After the hurricanes down in Beaumont, there were thousands of dead birds littering the ground. But, when you look at the video, you will see that the clouds themselves surrounding the twister are not spinning. This is because these are the dust and debris clouds surrounding the funnel itself. These clouds can extend out as far as a quarter mile, depending upon the size of the twister. These are the clounds you see the eagle diving into. Though, eagles can withstand flying in 120 mph+ winds. There are all kinds of crazy winds surrounding a twister and the thunderstorm that spawns it. Just as there are updrafts into the funnel, there are also downdrafts in the vicinity of the funnel also.
Thunderstorms can spawn all kinds of violent winds, and not just tornadoes. Hailstones are formed when raindrops are sucked up into the top of the storm, freeze, and then drop back down through the storm, only to be sucked up again, pick up more moisture, refreeze, and drop back down again. The size of the hailstones depends upon how many times they get sucked back up into the storm. Finally, they will get tossed out of the back of the storm. They can sometimes get thrown 20 miles or more, and land on the ground far away from the core of the storm itself.