An external plot is what we might think of as all the action that is directly visible. It sometimes can be best understood in a physical sense, or as a physical journey.
It is usually described using the terms inciting incident, super objective, obstacles, climax, and denouement.
For instance, in Harry Potter, the physical journey, or external plot, that runs through the series is that Harry must go to wizarding school, and ultimately that he must kill Voldemort because Voldemort is trying to kill him and it looks very much like one of them is going down hard.
Internal plot refers to the spiritual journey that the protagonist goes on due to the events of the external plot. It refers to spiritual growth.
It’s function is to flesh out the meaningfulness of the story. It will frequently help to answer the question ‘what was the point of all that?’
It makes the story appear richer, and more meaningful, and satisfying.
It is often constructed around the core question ‘Who am I?’
For instance, in the Lion King, according to Mufasa’s cloud ghost, Simba has forgotten who he is when he is living the carefree life of a Hakuna Matata in the forest with Timon and Pumba.
The events of the story teach him about his true nature and destiny. The external plot is that he must return to the pride-lands and kill his usurping evil uncle (Hamlet?), and the internal plot is one of spiritual growth towards self knowledge and community mindedness.
There can be internal plots that indicate an increased understanding of selfhood. Take Star Wars as an example. Luke Skywalker literally has no idea who he is at the beginning of the first film. The external plot is that he must defeat the evil emperor and restore balance to the galaxy.
The internal plot is one that is concerned with the revelation that Darth Vadar is in fact Luke’s father. Luke literally learns who he is. But this is also a journey towards self discovery about who he is morally, spiritually, and philosophically.
At the end of the film Luke decides he would rather die than take up the ways of the darkside, and save his own life and the lives of his friends in the rebel fleet. This is a journey of immense self discovery that results in Luke’s discovery that he is at his core good.
Writers will often focus first on the external plot and attempt to make it very compelling. Remember to bring heart and meaning to your story by including an internal plot, that is themed around personal development, and a movement towards spiritual growth.