What led to this trope?
The history of this trope's prominence in literature is long and complex, and dates as far back as the last millennium of ancient history. Since the first year of flight was 1903, for the longest time humans had a big fascination with the skies. What lied above the clouds? Was it possible to go up? In many religious texts, the realm of God (or gods, in polytheistic beliefs) is said to rest in the upmost heights, which a mortal can only reach in spirit after passing away. Even in the religious texts themselves, there are stories of people who attempted to reach the skies in order to access their Creator's whereabouts, to no success (one of the most famous cases is the civilization who attempted to build the Tower of Babel in the plain of Shinar). Over the centuries, many people were inspired by these recordings, which led to developments of all sorts (artistic, scientific and recreational) to learn more about the beautiful blue yonder. One of the earliest examples in fiction is the parodic book True History, published in the 2nd century by Lucian of Samosata and ofted nominated for the title of Ur-Example of Interplanetary Voyage. In the absence of airplanes and any sufficienly tall tower, Mountain Climbing was the earliest realistic way to travel up in real life, which inspired fiction based on the Scaling the Summit trope.
With the developments in physics and astronomy as of the 17th century, it was known that the celestial bodies visible during night lied well beyond the scope of the Earth's atmosphere, so our curiosity on the things we could see from the ground only strenghtened. For science fiction and high-fantasy literature, this provided an excellent opportunity to explore themes on exotic locations that could be found in the sky, if not beyond. One of the earliest works in this department was Somnium by Johannes Kepler, which tells about a man who dreams of traveling to the moon and finding a civilization there. Another work is Jack and the Beanstalk, which revolves around a boy who harvests a set of magical beans which grow overnight until reaching high enough for Jack to, after climbing, reach a castle above the clouds inhabited by a Giant and his wife. Another example is Gulliver's Travels by Jonathan Swift, which revolves around Gulliver encountering various exotic destinations along the way, the third being the floating city of Laputa. The book From the Earth to the Moon, written by Jules Verne and published in 1865, rose to fame due to its concept of traveling beyond the sky to reach a destination found in outer space (the moon); while it wasn't the first work of fiction where the moon was the final destination (it's beat out by many of the aforementioned sci-fi books, starting with True History); it served as one of the earliest serious, non-satirical executors of the idea, and gave inspiration for later sci-fi books which aimed to expand upon the themes of space travel, such as The First Men in the Moon by H. G. Wells.
During the 20th century, with the advent of aircraft invention (1903), the first succesful climb to the top of the highest mountain (Everest in 1953) and space travel (1969), what was the stuff of myths and mysteries in prior centuries reached a natural conclusion, but the trope is far from dead. In many literary and audiovisual media (books, film, television and video games), there are works whose stories revolve around characters undertaking a lenghty climb or flight to the skies, usually to meet an ulterior objective (and often executed by way of one of the associated Sub Tropes). In video games, for example, expect to do one if you're playing a high-fantasy RPG, platformer or action-adventure game. While the reason for the trope's prevalence in modern-day popular culture is not readily apparent, a possible reason has to do with the creator's interest in expanding the scope of their created world, since the realization that there's something up there allows the work to give new opportunities for Character Development.
Is it plausible in Real Life?
Well, it is obviously Truth in Television. Many of the tallest buildings in the world are called "skyscrapers" for a reason, even though none of them are higher than the lowest threshold for a mountain (the Burj Khalifa in Dubai is "only" 828 meters long): They're built with the idea in mind of presenting an attractive opportunity for tourists and workers who wish to have a nice, panoramic view from above the ground. Heading to the sky is also a recreational activity that comes in several branches of sports (parachuting, alpinism, flight exhibition, and so on). However, it's not something you'd want to do yourself unless you're a well-trained professional.
If you didn't sleep during the geography classes in school, then you'll recall that our planet's atmosphere is divided into different parts. The part that is generally agreed to be the closest domain of the sky to us is the upper half of the troposphere, which eventually borders with the stratosphere. The problem for a human with approaching this part is twofold: First, it becomes more difficult to breathe, because the concentration of oxygen molecules decreases as the altitude rises (you'd have to breathe harder and more often for your blood to receive as much oxygen as it does when you breathe normally at sea level); this is also attributed to the lower temperatures, which tend to greatly reduce a person's stamina. If it were possible to build a floating facility in the upper half of the troposphere, the humans inhabiting it would need to either have to develop a circulating system capable of working with the reduced breathable air (comparable to that of a mountain deer) or wear equipment with oxygen tanks.
Secondly, the closer you get to the stratosphere, the more exposed you'll be to the ozone layer; since the role of the layer is to protect the Earth's surface from ultraviolet radiation, oxygen here will become dangerous to you due to the radiation being absorbed by the molecules (the chemical composition of this oxygen won't be O2, but O3, which is ozone itself). Also, it may seem weird that both the upper troposphere and the mesosphere are extremely cold (the top of the latter is at -90°C, or -130°F, on average), yet the stratosphere is unusually hot; this is explained by the lower amount of air molecules to absorb the resulting heat from the UV radiation that migrates from the Sun. For this same reason, the atmospheric pressure decreases drastically, as the average ratio of weight-per-surface-unit is lower. Because most commercial airplanes travel within the lower half of the stratosphere, some people experience ear clogging (which can become painful) due to the major pressure changes to which the auditive system is sensitive. The air you're breathing inside the plane is actually trying to damp the effects of the lower pressure from the outside (and you can further alleviate your ears while chewing a gum), which is why opening one of the plane's doors during the middle of a flight is an easy way to meet a cruel death.
