What a mystic is
Mysticism is one of the three main professions available to exiles, along with fighting and healing. Whereas fighters and healers have fairly clearly defined roles, the role of the mystic is not as easily apparent. The mystic's specialty lies in knowledge, and many mystics are learned scholars. Mystics are also known, and sometimes criticized, for their extreme secrecy. It is this dichotomy of knowledge and secrecy that primarily defines the mystic class.
A common misconception that new exiles have about mystics, is that they think that mystics are mages or wizards, able to cast fireballs and shoot lightning bolts from their fingertips. Nothing could be further from the truth.
Most mystic abilities have little to do with combat at all; rather they focus on the areas of knowledge, communication, and transportation. A skilled mystic is able to gain knowledge about the world that other exiles cannot, such as where someone is fallen or the danger level of a nearby area. They can also foster communication by tuning sunstones, and teleport exiles across vast distances.
The one ability that mystics have that directly impacts combat, is the mystic "boost" which can temporarily improve one's accuracy or balance, helping to give a hunting party the edge in a difficult area.
How to become a mystic
Becoming a mystic is not an easy process, and involves solving mind-challenging puzzles, completing difficult quests, and gaining the approval of the mysterious Mystic Council. The path just to reach the first circle generally takes years, and many aspirants fail to become mystics at all.
For those not afraid of failure, the best way to learn more about becoming a mystic is to speak with one. Because mystics are very secretive, asking them how to become a mystic probably won't yield any concrete answers, but it will give one a better idea of what the class is about, and where to start. The rest is a journey that one must complete (for the most part) on one's own.