No doubt, prima facie what's most pertinent needs to be stated at the outset. There is an element of randomness to all air crashes.
That said, sustained efforts by US and some foreign airlines to identify and eliminate minor problems that generally are common precursors to accidents brought down fatality rate drastically from the '60s onwards.
One of the most common scenarios for a plane crash in the '50s was Controlled Flight Into Terrain(CFIT) referring to aircraft that were piloted into the ground, water, mountains or other terrain. However, improvements in cockpit instruments and jet engines dramatically improved aircraft performance and brought down the number of CFITs in the early '60s.
Look at the number of air crashes in 1959, 1960, 1961, 1962. The number of air crashes in the US fell from around sixteen in 1959 to around four in 1962. (Check above link for exact figures.)
Moreover, every single air crash that was successfully investigated provided data to the OEMs that was used to improve the aircraft. For example, the public inquiry into the Comet airliner disaster in 1954 found out that "Metal Fatigue" was the cause of the accident. The investigators found that a small weakness such as this would quickly deteriorate under pressure, and would rapidly lead to a sudden and general break-up of the fuselage.
Look at this image below. It reflects the causes of air accidents:
You will notice that some of the major causes of air accidents like Pilot Error and Mechanical Failure have gone down between the '50s and '60s.
Lessons were learnt from past accidents. Consequently, companies like Boeing brought about necessary improvements to the avionics and engines. Airliners also helped pilots hone their skills. A combination of these two factors brought down the number of air crashes.
Even to this day, OEMs and airlines are making effort to further improve flight safety.
One oft-cited example is a discovery in the last decade by US Airways (then US Air) that many of its planes approaching Charlotte/Douglas International Airport in North Carolina were coming in “high and hot” - too fast and at a steep angle. After the investigation into this issue was completed, the FAA changed the approach procedure there and the airport installed a system to guide planes at a proper angle.