Well, I haven't studied this or anything but it's pretty easy info to find and even a quick google explains it. The term game didn't become known as 'soccer' in the US, it became known as that in Britain and then that spread to the US.
There are lots of types of football of which Association football is just one. There is for example, Gaelic football, Rugby football, Australian Rules football, Gridiron football etc. The game of 'football' goes back to the middle ages, but this wouldn't have been much like the modern game and would have varied depending on the place. When football was codified in the 19th century it branched off into the various different types that I mentioned.
The name 'soccer' comes from the 'fad' in English universities of adding '-er' to the end of things, so you have rugby being called 'rugger' (and it still is, albeit jokingly, to this day) and since association football didn't adapt easily it became soccer.
There is more about when the usage changed in the article that I found on it -
http://ns.umich.edu/Releases/2014/June14/Its-football-not-soccer.pdf
At least 1911. Here is the New York Times article talking about when the brand new NCAA would recognize Soccer matches between schools: http://query.nytimes.com/mem/archive-free/pdf?res=9E06E0D91231E233A2575AC2A9649D946096D6CF
However in at that time it wasnt universal as the article uses, soccer, association football, and soccer football at various points.
However it was not of course a word invented in America, as the shortening of "Association Football" dates well before, here is as letter to the Oxford student body in 1885 referencing the playing of "Socker" :https://books.google.com/books?id=fPsHAAAAQAAJ&pg=PA171#v=onepage&q&f=false
So it is a combination of using the already existing name for the sport that was on common use in Britain "Soccer" and to further keep it clearly distinct from gridiron football which had developed at the same time from the 1870's on.
here are a few previous posts for more info
/u/metalbox69 gives a quick history on the development of the many football codes Can anybody explain when and how rugby, football, soccer, etc. developed?
more examples of slang using -er How legit is the theory that the word soccer was invented in the United Kingdom?
Just a point that "soccer" isn't an American thing particularly, but common where another football code dominates, e.g. Ireland (Gaelic football), Australia (Aussie rules football), Canada (Canadian and American gridiron football), as well as some other countries. Check out this map that gets posted on /r/mapporn now and then