The dark side that they are referring to is actually the shadow in Jungian ideas and is one of the major archetypes (elements in unconscious that pave the way for us in how to respond to things). The shadow is the place where we hold repressed memories and materials from consciousness. We are not directly aware of them but they are indirectly influencing our lives. It is not negative all the time as it can also store energy that bring about our creativeness, our essence. It is also the place where we hide our undeveloped negative and positive qualities.
In Luke’s case, he is constantly being asked to face the dark side by the emperor. Just like the shadow, the more he fights this, the more he refuses to tackle it on directly, he will suffer the consequences and not be able to win the fight or save his friends. He is torn with facing the shadow because in there, the step is to face on what is incompatible with the persona he has. In the end, he finally faces it and as he became aware of it, just like with Jung, he is not afraid anymore and it unleashes a great force of energy that made him more than capable of winning against Darth Vader.
In doing so, I can tell from the scene that Luke still has deep insight when it comes to his consciousness as he did not let the shadow overcome him, only derive energy from it. He knows that he is more than the shadow and if he were a patient of Jung’s… he would have done considerable progress in this area.