Much of this information comes from Superman Through The Ages' website. More info comes from the 31st Edition of The Overstreet Comic Book Price Guide.
The Golden Age officially started when Superman first appeared in 1938. The Silver Age officially began with DC's publication of the new Flash (Barry Allen) in Showcase #4, 1956. But there is no such distinct shift to a Silver Age Superman. There was an industrial change, led by the self-censoring Comics Code Authority, there was a rewriting of Superman's history coinciding with the restructuring of the editorial staff and the return of Jerry Siegel, and finally there was the attempt to explain the continuity of past DC stories.
The Golden Age was a time of great readership for comic books. In early 1946, the Atom Age began, which can be considered a subset of the Golden Age. This marked a time during which the industry hit their all-time highs with over a billion issues sold a year. By the early 1950s one in every three periodicals sold in the USA were comic books. Comics also had a wider range of genres during that time, with westerns, fantasy sci-fi (Flash Gordon), silly and satirical, horror, serious sci-fi, mystery, and mystical stories, and even super-hero comics. Comics were not just for kids (though super-hero comics pretty much were). They were the descendants of the old pulps. Adults read them. Girls read them. Normal kids read them. Not just fanboy geeks. Sadly, along with success came scrutiny, and suddenly comics were being blamed for destroying the youth of America. Congress actually held hearings about the dangers of comics to the children of America. Frightened by such scrutiny, the comics industry leaders quickly developed their own form of censorship to assure parents that comics would be safe. They developed the Comics Code Authority, which had a strict list of requirements necessary to receive the Comics Come approval. Examples of their requirements are as follows: Illegal actions must always result in suffering and punishment by the end of the issue (meaning no getting away with robbery until the next issue); Drugs could not be depicted as not having negative effects (so even your first time using a drug, you must suffer as a result, making realistic anti-drug comics impossible to print); Comic book characters must speak in proper English, using proper grammar (no slang, no realistic dialogue). These numerous requirements took much of the literary quality out of comic book stories, resulting in rather childish, unrealistic stories, in which major social commentary could not be well presented. This drop in story quality caused many of the genres no disappear. Super-hero comics ended up being the main surviving genre, but even they took a huge hit in readership as the characters were getting stale. It was at this point that DC decided to dump their previous history and recreate their characters in completely new forms. They drastically redesigned Flash, Green Lantern, Hawkman, Atom, and others. Superman, Batman, and Wonder Woman are some of the few characters to survive this revamp without any major changes. But the Golden Age was over, this is now the Silver Age of comics.
Leading up to the industry changes into the Silver Age, Superman had already undergone many changes. Following the departure of Siegel and Shuster from DC in 1947, Superman was taken in a vastly different direction than before. Previously being a character who considered himself to be thoroughly human, suddenly Superman started to take an intense interest in his Kryptonian heritage, and so did the writers. It was during this time, 1948 through 1958, that the Superman tales shifted to being heavily fantasy sci-fi stories (some, like Jim Steranko, consider this to be the decade which made Superman into a laughing stock with its silly stories). Along with this new attitude in the Superman stories, came the first major continuity hole in Superman's history. Though it had been well established that Clark never donned his suit until he became Superman in Metropolis, suddenly Superboy showed up in More Fun #101 (1945). This also marks the first time Kryptonians are depicted as not having super powers on Krypton, totally revamping the origin of Superman's powers (finally attributing his powers to our lesser gravity, but NOT the yellow sun). Despite these facts being established, Superman stories continued to ignore the existence of Superboy. It wasn't until 1951 that Superman stories finally acknowledged the past 6 years of Superboy stories as being a true part of Superman's continuity. It wasn't until 1949 that Superman even knew he was an alien (after the discovery of kryptonite). Finally, in 1959, the Superman character solidified into the Silver Age version. First, in the late 1950s, sole editorial control over Superman was given to one person, Mort Weisinger. Second, in 1959, Jerry Siegel returned to DC to write Superman. It was at this point that Superman's history as Superboy was completely assimilated into the Superman mythos. Actually, the writers threw out all of the old Superboy stories, but they retold the same basic stories to create a new continuity. Finally, we have a complete version of the Silver Age Superman.
With the Silver Age in full swing, we now have this new version of Superman fighting alongside the new DC characters, and the Justice League of America. But DC did not tell the readers that this was a new Superman like they did with Flash, Green Lantern, and others. This is apparently the same guy who fought alongside the Justice Society, but the JSA apparently never existed. The fans insisted on an explanation, and DC provided it. It was at this point that DC took the interesting approach which makes it unique in the literary field. While most literature just ignores old stories which no longer fit in continuity, DC took the unique step to explain why old stories are ignored, effectively making them a part of continuity without having to acknowledge them. DC decided to introduce the idea of multiple realities, or in particular, multiple Earths. It turns out that Earth-2 was the world in which the Golden Age stories took place, with heroes active during World War II. In this world, Superman first appeared in 1938 and never acted as Superboy. In this world, the Golden Age existed and continues to exist. These new stories, these new heroes, all fight together on a different Earth, Earth-1. Superman is one of a few characters who actually exists in both realities (along with Batman and Wonder Woman). On Earth-2, stories were allowed to follow the heroes as they aged, some died, some grew up. Earth-1, however, kept the heroes at a mostly constant age.
With all of these factors combined, you can see how the Superman character initially was revamped into a Silver Age character, and then how the original Superman became the Superman of Earth-2. This method DC used, like Crisis much later, may not be all that well liked by some readers, but is a unique way to deal with continuity issues as times and characters evolve over the years.
