The explanation of man becoming something more didn't really "start" with advanced scientific phenomenon; in truth, it was an occurrence from ancient times. There is most usually some type of transformative occurrence which leads to the metamorphosis. This has taken many, many forms over the millennia,from:
- a chance meeting with higher beings and abilities being "bestowed" (parallels between the Gilgamesh and the gods and Billy Batsom and Shazam)
- to attaining powerful items (King Arthur and Excalibur to Dark Hawk finding his amulet)
- Powers being somehow connected to nature (Odysseus getting the bag of winds to Alec Holland becoming Swamp Thing)
- Abilities being the result of some incredible, unlikely circumstance (from Achilles being dipped in the river Styxx as a baby to Peter PArker being bitten by a radiocative spider)
- To gaining abilities through training, trial and development ( from the Monkey King in Journey to the West to his modern day equivalent, Goku, or even Batman.)
- Potions and elixirs (Alice in Wonderland to Captain America's Super soldier serum...actually, there are a LOT of magical potions/super serums out there)
- to heroes being born with their abilities and meant for greatness (Hercules to Superman and the various X-Men.)
The endowment of extraordinary abilities from some form of communing with nature is a very common tread as well. From Buddha reaching enlightenment through meditation and spiritual elevation, to people becoming werewolves and vampires via a bite (parallel to Spiderman, here), it's been almost a trope that supernatural powers somehow extend out of nature, or rather, it has a strong link throughout legend.
The Science fiction aspect of it most probably started with Mary Shelly's Frankenstien. It became more prominent in the 1930s post pulp era, growing out of the Industrial Revolutions's explosion of scientific discovery and arguably really took off as new scientific advancement began to gain ground in the 40s and 50s. This supplanted pure "nature" as underlying factor, because now mankind had the ability to viably, or at least speculatively, manipulate nature itself to lead to some otherworldly or unnatural effects. This was yet another reason many movies of that 20 year period took a fantastical, Sci-Fi bent.
"Super powers" being a product of such mysterious, yet speculatively plausible scenarios, is somewhat of a natural byproduct. In the early 20th century, radiation and its scope was a little known-about wonder; by the mid century, harnessing the power of the atom was a hard scientific fact. The X-Men were seen as the "Children of the atom" because it was shown that radiation could in fact cause mutations in DNA. In the 21st century, DNA manipulation, from CRISPR to "bio-hacking," to produce various effects is a very real phenomenon, with examples of it seeming to make humans "better" than our previous limitations. As such, it's not hard to see where a lot of this type of thought came from, and why as our knowledge as a species continues to grow, it continues to be such a strongly held perspective.