@JTam This goes back in time a bit, so excuse me for the longer explanation. In 2001 when I went through boot camp, the Marines still did the KD course of fire followed by a "field fire" portion that had you wearing flak and Kevlar at a known distance. There was a section of that where you shot with a gasmask, but in general that was the most you got in many units. So the Marines decided to up the game, and introduced the EMP (Enhance Marksmanship Program) to bring their rifle standards forward.
The EMp introduced more advanced shooting techniques such as firing from the hip, shooting without sights, shooting on the move, etc. Until the EMp program, the basics were Gospel, and shooting on the move was "I'm up, they see me, I'm down." Part of the instruction for the EMP shooting on the move was using "THe Groucho" to move about, which helped maintain a stable firing platform while allowing you to move.
The reason it was called "The Groucho" was because it looked like this guy- Groucho Marx.
One of the instructional cues was to walk like Groucho. The other was to think of sitting on a barstool, and walking toe to heel while rooling the foot. If you've ever seen the Marx brothers, you'll notice that Groucho sometimes walks like this- hence the instructional cue. Groucho was a simple term to use, and sounded cooler (I'm guessing) than "Bar Stool Walking" which sounds like a 70's song.
"And everybody was Bar Stool Walking!!! They were shooting as fast as lightning!!! Putting holes into the badguys!!! Two to the chest and one between the eyes!!!"
In summation- it was meant ot allow the shooter to engage targets in close quarters where standard rushing was impractical. Groucho was an instructional cue that we adopted as the name of the technique, though we never took the opportunity to put on the fake mustache-glasses things and chew a cigar while we did it.