Originally, this expression was meant to denote one figuratively brandishing their genitalia in an effort to convey their excitement. Paradoxically, the genesis of this expression was almost responsible for the demise of it. "Get your burger out" was an abysmal and awkward failure at launch, partly because it's kind of not that funny but mostly because it was tyrannically thrusted upon the public lexicon. Furthermore, one could hardly ignore the contrivance of how forced this expression was when it was supposed to define a bit of spontaneity.
Peter: "Just scored some tickets to the all cat opera! Get your burger out!"
Amanda: "What do hamburgers have to do with anything?"