The word began, as near as anyone can tell, when the absurdist Twitter account BirdsRightsActivist tweeted the single word “Birb,” out on November 2012; two years later, it had multiple entries in Urban Dictionary and a dedicated reddit forum.
There are other similarly wonky misspellings such as smol, snek, and chonky. It’s not meant to be an abbreviation; it’s meant to be slightly off-kilter.
BTW, there’s a book out titled Because Internet that covers how the Internet shapes language. I haven’t read it myself yet, but it sounds like a good read.