Nimrod is actually a biblical figure. He was a king that appeared in the books of Genesis and Chronicles. He is referenced in the Bible as a "mighty hunter before God." Since then, he had been used as the ultimate comparison for a mighty hunter. And then Bugs Bunny happened.
In one of the Looney Tunes cartoons, Bugs Bunny refers to Elmer Fudd as 'a nimrod.' He was making a sarcastic comparison of Fudd's abilities to the ones of the great biblical hunter. Year later, the term changed in American English to be a derogatory term, and a synonym for idiot, or stupid.
Bugs Bunny might've popularized the usage of nimrod as a synonym, but he might've not been the first to use it as such. The Oxford English Dictionary traces the first usage of nimrod in a sarcastic / derogatory way to a 1933 writing that predates Bugs Bunny's by 5 years. Either way, it's interesting that a cartoon character can have so much influence, not just in pop culture but in language as well!