I presume you mean this only related to film, as psychological horror had been around for much longer in literature.
As far as film is concerned, it's clear the horrors of both World War II, followed by the paranoia and fear brought about by the cold war played a huge part in the rise of the genre in film. The possibility of nuclear holocaust coupled with the waves of fear spurned by McCarthyism and other fear movements of the era are clearly influential. There are the horror flicks themselves, and also elements of psych-horror in other genres, as well. Consider the late-cycle film noir Kiss Me Deadly, with its secret case that contains a mysterious glowing object that burns whomever goes near it. The contents of the case are radioactive, burning and searing whomever opens the case, leaving them forever scarred. Clearly based around the fear of nuclear holocaust that was running rampant at the time.
So, not only did the actual genre of psychological horror flicks emerge during that time, the concepts spilled out into other genres, as well. In this case, a private detective flick. If the image looks familiar, that's because other, more recent filmmakers have paid homage to this film, and that shot in particular, in films like
Raiders of the Lost Ark, Repo Man, and Pulp Fiction.
So, not only can you pinpoint the exact reasons for the movement, but also, one can track the influence across the years as various films from subsequent decades reveal influences and concepts that clearly link back to this era.
So, there you have it: I cite the two biggest influences in Psychological Horror in film to be
World War 2 and
The Cold War.