Meaning/Usage: To not know; to not understand 不懂 不知道
Explanation: Not many resources on this phrase, but could come from beating information out of someone. When someone is holding back information, they would get beat until they tell the answer. In this case, the person could be saying that even if I was beaten, I wouldn't know. Therefore, it became popular to say "beats me" to indicate you do not know.
"It beats me how they finished before us."
"Beats me why he was so mean to you."
"Beats me how she found out we did it."
A. "So how did you and Tim do in your race?"
B. "I finished in third and Tim finished in second."
A. "You got third? How did Tim beat you?"
B. "Beats me, I guess he has been training really hard."
Other Common Sentences
"I have no idea how to do this."
"I don't know what he's talking about."