Radiation was not well understood, and as such either scary or promising - or both, depending on whom you asked. It's little wonder that fiction turned to it for experimentation. For instance, Lori Maguire writes that:

The late 1950s saw the arrival of a host of new villains, most of them showing the influence of science fiction.

[...]

An analysis of these figures and their presentation can reveal a great deal about the anxieties of the time. In particular, three themes stand out (although there is some overlapping between them): fear of nuclear radiation, fear of invasion, and fears related to the space race. Underlying each, of course, is feat of technology being turned to the wrong ends and used to take away freedom and establish dictatorship or, quite simply, to destroy.^1

Maguire's article is about the fear of technology manifesting itself in Superman comics, meaning that it mainly concerns the ethically negative power of radiation - and also, by extension, that even Superman is actually vulnerable to a radiating substance: kryptonite, which forms a duality of his origins.

The same basic reasoning was behind other, similar powers as well, though. Gresh and Weinberg expand on this nicely:

No one, including many scientists, was sure of the effects of hard radiation on animal tissue, and dozens of B-movies and comics featured radioactive monsters. Nor were people sure of the effects of radiation on the human brain.

The Hulk took paranoia about the bomb to the next level. The Hulk was the terrible by-product of atomic bomb radiation. Making things worse, he was both a good man and a horrifying monster sharing the same body. The concept of dual personalities in the same body was already a cliché, but it served Lee and Kirby well. The Incredible Hulk was blood brother to the Amazing Colossal Man and nephew to Dr. Jekyll and Mr. Hyde. The Hulk was a rampaging monster and an innocent victim combined in one.^2

A great article on making animals huge in early modern cinematography was written by William M. Tsutsui^3 and is hard to summarise succinctly, as the author goes through a number of interpretations, fears, cultural differences, etc., in a short amount of pages. The author starts at the same point as above:

The oldest and most esteemed critical approach to movies like Them! is that they are all about nuclear fear, the widespread anxiety about the threat of atomic annihilation that (so the story goes) gnawed at the middle-class psyche throughout the glory days of the Pax Americana.

The inherent dualistic nature of this fear is also covered, as are - more importantly - criticisms of this theory, namely that just because there are huge monsters in a movie, that doesn't mean we have to overanalyse ourselves, nor the filmmakers. Tsutsui quotes Bill Warren, from an 1982 article that I haven't read, but the quote makes a very clear point:

"In the 1930s the equivalent gimmick was electricity; in the 1920s, it was surgery and often gland operations. In the 1950s, it was radiation that got the monster going. The 1930s didn’t suffer from fear of electrical annihilation; although the 1950s did tend to be worried about atomic warfare, radiation in science fiction films wasn’t a means of expressing this fear, probably not even unconsciously. It was just a way of originating an unusual or interesting menace. ... Radiation was used to explain many wonderful things, from giant insects to walking trees to resurrecting the dead. This was not a form of nuclear paranoia, merely cheap and simple plotting."

Moving away from the subject of radiation, Tsutsui further discusses Cold War aspects and the Freudian interpretations of giant insect films. He also introduces his own theory about humanity's fear of insect infestation, that was quite prevalent at the time, being the driving force behind this development. You may find this interesting, but it's got to do more with why we had giant insects than why they turned big through radiation.

^1 Maguire, Lori. "Supervillains and Cold War Tensions in the 1950s." The Ages of Superman: Essays on the Man of Steel in Changing Times (2012): 16.

^2 Gresh, Lois H., and Robert Weinberg. The Science of Superheroes. J. Wiley, 2002.

^3 Tsutsui, W. M. (2007). Looking straight at THEM! Understanding the Big Bug Bovies of the 1950s. Environmental History, 12(2), 237-253.