When something happens preternaturally, it's extremely unusual — you could even say it's strange and mysterious. If your dog is preternaturally smart, you may feel like she understands every word you say.
This adverb is commonly used to describe things that happen in a bizarre or extraordinary way, particularly if they're hard to explain. A magician may cause your friend to appear to hover preternaturally above the ground, for example. Sometimes its use signals that you can't quite believe your eyes: "Those redwood trees were preternaturally enormous!" The Latin root is praeter naturam, "beyond nature."