The classical transformations of a vampire are threefold. First is the iconic bat. It’s easy to understand the connection as bats are nocturnal and a species of bat is vampiric. Enough said. The less we dwell on ideas such as the conservation of matter, the better.
Next we have the mist. Again, this is understandable as the regions of Hungary and Romania—especially near the mountains—frequently are shrouded in fog. It’s not difficult to imagine that the concealing mist might be of supernatural origin, and would explain a vampire’s ability to appear and disappear into the night.
Lastly, and it’s not completely canonical, is the wolf. Again, geographically this animal is perfectly at home in the mountains and forests of Hungary and Romania, and its howls into the night speak of a dangerous predator that warns people to be afraid of the night.
The etiology of this aspect of vampires is unclear, and the transformations serve little real purpose for the vampires themselves. Why would the vampire need to transform into something else? What’s the point? A bat gives it the ability to fly, possibly moving about stealthily, but so what? The same is true with the mist, though mist is rarely able to move about freely or with any kind of speed. Wolves would seem to attract more attention than anything.
I would venture that the transformations are a way to instill more fear into people. The three forms only have that in common, particularly in the medieval minds that first dreamt up the vampires. but this trait has become popular, and is one of the more magical aspects of their nature, ranking up there with their immortality.
As a purely magical trait, it serves to further distance them from humanity, giving rise to Satanic influences—quite possibly popularized by various Christian sects—as a warning to people. Regardless, it’s a fascinating trait, revealing more complexity to an already complex mythology.