Built into the app is a feature called Cinema, which acts as a gyroscope that continuously crops the external frame around an anchored a central image. The edges of the fixed white frame never cross the warped, shaky frame, meaning when the images are collated together, all of the effects of camera shake are removed:
Instagram yesterday outlined how this neat algorithm works. The gyroscopic samples and frames are first fed into a stabilizer to generate a new set of camera orientations as an output. Each input frame is changed by the IGTrackStabilizer until it is in the right camera orientation.
Cinema also has an adaptive zoom feature, which picks the zoom of the camera based on how shaky the recorded video is. If there is a small amount of shake, the camera will zoom in more - as not much room is needed to counteract the shake. If there is a lot of shake, the camera will zoom in less, as a lot of space will be needed to make sure the shake doesn't mar the picture in the central frame.
The end product is smooth video playback, regardless of whether you are able to hold a camera steady or not.