It is said that the emperor of China gave this surname to a commoner out of gratitude. He was given this surname because he lived in hiding in the forest for five years.
The character "林" with two trees actually means a grove or a copse. (A single tree is "木") If you looked up "forest" in a Chinese dictionary, the character you would get is "森" with three trees. 木, 林, 森. Tree, grove, forest.
Yes, it can be a variant of Lynn, usually. It can also be a nickname for Linzee, Lindsay, Lindsey, Linda, and so on. It's very beautiful. I'm going to name my daughter in the future Linzee, nickname Lin.
(I'm Chinese and this is part of my name).