As I understand it, a Live Photo is a hybrid of video and a photo. The video is probably much lower resolution than the core "still photo" so when you see the final photo, you are seeing the high resolution still instead of a lower resolution video frames. It's not that it's necessarily brighter but that there is much more detail to see. If you think of pixels as tiny little lightbulbs to turn off or off, more bulbs are available at megapixel resolution for stills vs. even 4K or 1080p video frame resolution for video (I don't know if Live Photos uses 1080p or 4K for the video portion).
You can likely see the same effect by taking a still photo of something and then switching to video and taking some video of the same thing. Then compare the "brightness" of both.