Being full of beans can either mean you're energetic, lively, and brightening up the room, or it can mean you're talking nonsense.
There's a good chance that full of beans meaning "full of nonsense" is a softened version of other, more swear-filled phrases meaning the same thing. But the version that means "full of energy" is a little more mysterious. You could think of it like this: beans, when eaten, give you long-lasting energy. If you're full of them, you've probably got plenty of juice for running around and doing activities.