It was an established staple of the science fiction background, so much so that if you wrote a story set in the future that did not have overpopulation, you had to explain to the readers some reaslistic reason why, in much the same way if you had a future where there was no interplanetary travel, you would have to mention a reason: Otherwise the readers would not find the tale believable.
Why is it so persistent a fear?
That is not hard to see: the writers in the 60′s and 50′s were born and raised in the 30′s and 20′s, and they lived through urbanization, industrialization, and the postwar Baby Boom. It also means that read stories and heard yarns from the previous generation of writers in the 1890′s and 1880′s. This includes the time when the open ranges of the West were still in the process of being fenced in an closed down — the frontier was filled up and closed, the manifest destiny was complete, and the frontier spirit was dying off. They saw highways and factories and parkinglots going up on land that used to be forest where their brothers and fathers had spent time hunting and fishing.
Nothing was more natural but that the science fiction writers would extrapolate from their current circumstances and foretell tales of Malthusian overpopulation.