I think Jesus was speaking of the physical, moral and spiritual aspects of man's personality.
All of us are basically animals but we have risen from being just animals to being moral creatures struggling to live decent lives in harmony with others in peace and with justice. Some of us look for deeper meaning and search for more. Some recognise a spirituality and see a creator as the origin of that and religion as a means to reaching understanding of it.
Jesus teachings leads us to God bringing us through these three stages.
Or perhaps that is a reflection of the principle of everything coming in threes.