For no particular reason, this story is set in the stone age, millions and billions and trillions of years ago, probably before some of you were even born.
Casper (a caveman) and Fido (an apatosaurus) go duck hunting and find Daffy. Casper slingshots a rock at Daffy, but Daffy manages to avoid it by disguising himself as a traffic cop. When the rock realizes that it has been tricked, it backtracks towards Daffy but ends up hitting Fido. Fido proceeds to perform a dazed dance.
Daffy snatches Casper's slingshot and tricks Casper into thinking that swimming is not allowed to prevent him from pursuing him. Subsequently, Casper and Fido leave, but Daffy, knowing that Casper won't give up, paints himself on a nearby stone. Casper, holding a stone club, sees the painting and bashes it, but the force backfires and makes Casper dizzy. Daffy gives Casper a glass of water, which cures the dizziness and earns him Casper's trust. Daffy, however, gives Casper a card advertising a rare, gigantic duck living nearby, which Casper and Fido begin to hunt, following billboards (parodying advertising techniques of the 1930s) planted by Daffy. They eventually reach the giant duck inflatable balloon pumped up by Daffy, terrifying Casper until Daffy gives Casper a knife with which to stab the duck. Casper does so, and the ensuing explosion kills them all.
The short ends showing the three sitting on clouds. Fido plays a harp while Daffy and Casper lament their mistakes (Daffy: "You know, maybe that wasn't such a hot idea after all!" Casper: "Good night, folks").
Most of Chuck Jones-directed cartoons from this era (such as the ones featuring Sniffles the Mouse), were very heavily inspired by Walt Disney's cartoon shorts, placing more emphasis on story and animation than on gags. Daffy Duck and the Dinosaur shows the faintest hints of deviation from such cartoons, which would eventually lead to the fast-paced Jones cartoons of the 1940s, such as The Dover Boys and The Draft Horse.
This is also an important milestone in the evolution of Daffy Duck's personality. While Tex Avery and Bob Clampett had depicted Daffy as completely insane, irrational, and uncontrollable in their previous cartoons with the character, Jones depicted Daffy here as somewhat more thoughtful and calculating. Chuck Jones and Friz Freleng would continue to develop Daffy's personality in this direction throughout the 1940s and 1950s.