There are two reasons I will offer for the demise of the rigid airship. Firstly, the 1937 Hindenburg disaster at Lakehurst, New Jersey, had the misfortune of being the first disaster ever recorded for mass media. But secondly, and perhaps more importantly, the Zeppelin Company in Germany did not survive the outbreak of World War II as its airships were of little military value once the war in Europe began.
Before this event, disasters such as the sinking of the ocean liner Titanic or the zeppelin Akron had to rely on eye-witness accounts. But the Hindenburg explosion and immediate aftermath was caught on both film and audio with a reporter, Herb Morrison, narrating the event as his composure broke. Within a day the pictures of the zeppelin's demise were plastered on front pages, within a week the film was being shown on newsreels all across the country, and within months across the world.
The other airship tragedies were primarily military (WW1) and the postwar loss of the USS Shenandoah, USS Macon, and USS Akron. All three of which were lost in the twelve years previous to the Hindenburg but with only eye-witness accounts and wreckage, when circumstances allowed.
The timing of this could not have been worse for the German Zeppelin company run by Dr. Hugo Eckener. This was the only company in the world serious about rigid airships for commercial use, and they had been working on it successfully for decades. Dr. Eckener was aware of the rapid advances in airplanes (the first non-stop transatlantic commercial flight was only a year later in 1938) and was looking to secure a future for the company as first-class rapid travel with luxurious accommodations. The Zeppelin Company had an outstanding safety record prior to the loss of the Hindenburg. But with millions of people able to watch, and re-watch, the Hindenburg bursting into flames and crashing there seemed little future left for the commercial airship.
All was not lost. The Zeppelin Company was already building a sister ship to Hindenburg, the Graf Zeppelin II, and reduced its passenger size so to fly with helium gas instead of hydrogen. The Zeppelin had long been a popular symbol for the German public, and there was much support to take back to the skies. However the United States had a monopoly on the world's helium supply from the Amarillo natural gas fields, deemed it a strategic resource, and halted exports to Germany with the 1938 Nazi invasion of Austria. Later that year, the Graf Zeppelin II was conscripted by the German Air Ministry for reconnaissance flights around Great Britain. But when World War II formally broke out, the two remaining zeppelins (the famous Graf Zeppelin I and the larger II) were scrapped and their massive hangers were demolished. Dr. Eckener, who had led the development and dream of the zeppelins, left the company and moved on. The last commander of a passenger zeppelin, Max Pruss, later declared before his death, "It was not the catastrophe of Lakehurst which destroyed the Zeppelin, it was the war."
"Today in Media History: Hindenburg explodes." David Shedden, Poynter Institute, 2015.
"Dr Eckener's Dream Machine: The Historic Saga of the Round-the-World Zeppelin", Douglas Botting, HarperCollins, 2001.
Also, please refer to this excellent answer by /u/Noble_Devil_Boruta