Why did cars stop being made with tailfins?

Style. The last car with fins was Cadillac & the first.

They always looked stupid & eventually enough people realised this & they faded out

about the mid 60's , they became unfashionable and expensive to make the fins .

Whale-fins was the new style  sometimes known as rear spoilers.  It was a "style" for a couple of years- not on every car.  55-57 Chev,FORD,Dodge did not have fins. Chrysler and Buick did up until 62? Mercury had them too in those years.  VW never had them.  Farm tractors and trucks never needed them.  Running out of cars back then.

Many of automotive press and much of the public were getting weary of the exaggerated tailfins, and the manufacturers were ready to phase them out because they added cost and complexity to design and manufacturing. Tailfins descended throughout the early 1960s, even adopting a downward slope on the 1965 Cadillacs.

They looked good up until recent body designs and most older vehicles are RWD but have entirely switched to FWD and AWD. Even the little eco cars like a corolla where once RWD.

For one thing tastes change and into the mid-sixties fins were seen on cars because of the pursuit of speeds in excess of the speed of sound and the space race.

I still have a soft spot in my eye sight for the cantered fins on the 57 Bird with the "afterburner" tail light.

Styling took it ques from the aerospace industry.

Not that the idea is dead, with front splitter plates, rear wings, rear diffusers and the like, cars have a lot of design ques taken from modern racing. Air flow is now more of a science then back when the fins were tall. 

Chevrolet started the no fins trend in 1961 and the rest followed.

It was a style.  People eventually realized it was silly.

The tailfin look was very popular during the early space age, making cars look like rocket ships or jet planes gave the impression they were very fast, but that makes no good sense, in reality the fins give no advantage in speed, stability or economy when they are tested using modern methods, now its considered an outdated retro look, a bit like wearing fashion from the 1950s, also there were safety issues, in a rear end collision the chance of serious injury was higher with sharp protruding bits of steel