This thing with the Commanders started in the early 1970s when George Allen arrived in Washington.
Allen, the football coaching version of Richard Nixon (complete with the paranoia and the hand-wringing), was obsessed with the Cowboys.
NFL Films has video of an interview with Allen talking about the Cowboys and he said, "The Cowboys had this division to themselves for so long. We came in and destroyed their power and that's what made the rivalry."
I'm paraphrasing for the most part, but the part about "destroying their power" is a direct quote.
If disliking Allen wasn't enough, he put together a roster chock full of people that inspired dislike. Guys like Diron Talbert, a journeyman lineman who made it his life's goal to hurt Roger Staubach; Billy Kilmer and Sonny Jurgensen, a pair of beer-bellied, a-hole quarterbacks; later came people like Joe Theezeman; Daniel Snyder, Art, the Fun Bunch, the Hogs, Spurrier (lol), etc.
Dallas has more wins than Washington in the rivalry, more Super Bowls, more fans, more class, more arrogance (but that's ok
), better-looking uniforms, better-looking cheerleaders, etc.
What it boils down to is envy.
From George Allen's obsession with Landry and his team in the early 1970s, to Daniel Snyder's obsession with Jerry Jones and his franchise now, that's what it's all about. Pure-dee, burgundy-colored jealousy.
Hail to the sub-par, second-tier, racial slur, wanna-bes.