All i can offer here is an unworthy and simplified explanation:
Hesychasm is basically a practice of "inner stillness" where a person, through prayer, enters the depths of their heart to commune with God in unceasing prayer. Through this communion, a person is purified of the effects of sin which has caused man to become spiritually blind and unable to perceive God. Once man reaches such a state a purification, it has been known for them to behold the uncreated light...
The uncreated light that is beheld by the hesychast is the same light that Peter, James and John beheld on Mt. Tabor when Christ was transfigured. The church has taught that the light the apostles saw on Mt. Tabor was not a created light that is seen with the eyes alone. Instead, it was at that point that they were allowed to see Christ's glory. This is also related to the beatitude "Blessed are the pure in heart, for they shall see God."
This is an extremely basic and unworthy explanation, so forgive me. And I am sure that if I have erred on anything, it will be corrected!
Christ be with you!