By twisting their three necks around like the blades of a helicopter.
That’s … that’s not how they fly, Bill. —LH
By jumping really, really high.
Keep trying. —LH
Magic!
Okay, seriously, no. —LH
Originally, they couldn’t. Then some asshat from game freak decided somewhere along they could.
Game Freak? Are you referring to the gaming company headquartered in Celadon City? I can’t imagine what they would have to do with the dodrio’s physiology or physical capabilities. Usually, it’s either Silph Co. or the Pokémon Laboratory who’s responsible for the creation and modification of pokémon for no apparent reason other than to see if they could.
The Pokédex According to Bill 