I don't have enough information to determine who Bugsy is. It could be a fictional character created specifically for this 'bugsy out of bedtime stories' concept.
It's possible that Bugsy is a wise character. Despite being part of bedtime stories, he could be the one who imparts valuable lessons. For instance, he might teach the other characters (and the children listening to the story) about the importance of sharing, or how to be brave in the face of fear. He could be an old, experienced creature in the fictional world of these stories.
Not having read the story, Bugsy could be a variety of things. Maybe Bugsy is a small insect with a big personality. He might be the kind of character that children can easily relate to, someone who has little adventures during the day and then comes home to a warm and cozy bed at night in the story.
Well, without more context, it's difficult to say exactly who Bugsy from Bedtime Stories is. It might be a made - up character like a little bug or an animal that has adventures during the bedtime stories. Maybe it's a character that helps teach kids a moral lesson in those stories.
Bugsy might be a unique character in the bedtime story. He could be a brave little being. For example, if it's a story set in a forest, Bugsy could be a squirrel who is known for his daring adventures. He might be the one who always explores the farthest parts of the forest, looking for the biggest acorns or the most beautiful flowers. Or he could be a friendly bug that helps other forest creatures when they are in trouble.
I'm not entirely sure as the statement is rather unclear. 'bugsy' could be a name or something specific in a certain context, and 'off of bedtime stories' is a strange construction. It might mean something related to Bugsy being removed from or not being part of bedtime stories. But without more context, it's hard to be precise.