@futurebird This appears to be one of a series of "I'M THE GUY" pins from 1910s cigarette advertisements. If you google "I'M THE GUY 1910s pins", you can see a bunch of cheesy, and often nonsensical ("put the air in hairpins"?) wordplay like this.
My best guess: ants are small, giants are big, by taking the "ants out the giants" you're the guy that took the small out of the giants, and that's why they're big. Of course, it could also just be nonsense.