The image is of a monkey swinging from branch to branch. This is an excellent representation of consciousness, which jumps restlessly from object to object. The karma of our past deeds determines what object consciousness alights upon, just as the momentum of the monkey's leap determines the direction of his travel.
This process continues without pause, repeating itself moment after moment throughout each beings life. In the context of the dependent origination, it refers especially to the moment of birth, called rebirth-linking consciousness, when the past karma of the being drives it on to a fresh becoming in this or that order of beings.
The image of the monkey should not be taken too literally. There is in reality no thing which carries on from moment to moment, let alone from life to life. There is only conciousness arising again and again afresh to a kaleidescope of shifting objects. The continuity we perceive is an artifact of the inertia of karma, imposing patterns on experience.