I think the Pokedex explains this well
From Black and White: "Because its genetic makeup is irregular, it quickly changes its form due to a variety of causes."
Pretty much all pokedex entries state that Eevee's abnormal genetic code gives it the ability to have far more potential evolutions than average. It is also highly adaptable, befitting of this capability, and in addition has the ability Adaptability.
Basically, Eevee is an abnormal Pokemon genetically whose adaptability allows it to take on many forms
Hope this Helps