Although neither Bella nor the reader can possibly know the answer to this question at this point in the story, we do find out the actual answer in chapter 14. By this point in the story, Bella knows that Edward and the other members of the Cullen family are all vampires. She is incredibly curious about them, their habits, and how they came to be. As the conversation continues, Bella notices that Edward knows more about her house than he should. She finds out that Edward has "spied" on her at night. Bella is taken aback for a moment by this information, but she is more curious than anything else. Bella asks Edward how often he visits her at night while she sleeps, and Edward confesses that he comes just about every night. This confirms for readers that Bella more than likely really saw Edward in her room earlier.
I still didn't turn around. "How often did you come here?"
"I come here almost every night."
I whirled, stunned. "Why?"
"You're interesting when you sleep." He spoke matter-of-factly. "You talk."