After doing my own research I found that the apple pie was invented in England in the 1300's. With that being the case, why do we often hear the saying "It's as American as the apple pie"? Because I had always thought the saying came from America inventing a very tasty desert, but apparently America didn't even invent it. Why do Americans claim it? Is there something I am missing?
Just because a nation claims a cultural icon doesn't mean they necessarily invented it; the Statue of Liberty for example is made by the French, depicts a Roman goddess, and is mostly known for its links to immigration.
Apple pie is, as you've noted, very old and well known in Europe centuries before USA existed. Apples were however very important to early European settlers, being mostly used for cider (see Johnny Appleseed). Thereafter apple pies became popular but it wasn't closely associated with the United States yet.
Things started to change around the turn of the 20th century; in an editorial in The New York Times (May 3, 1902):
Pie is the American synonym of prosperity and its varying contents the calendar of the changing seasons. Pie is the food of the heroic. No pie-eating people can ever be permanently vanquished. ...
In our own glad and fortunate country the seasons are known by their respective dominant pies—for each there is an appropriate pie, with apple pie for all the year 'round.
While the article is about pies in general, and names mince, custard, lemon, rhubarb, berry, peach and pumpkin pies, apple pie is named first and noted as the only one available "all the year 'round". This was made possible by preservation techniques developed by Pennsylvania Dutch pioneers.
It was only until WWII that "as American as apple pie" really took off, where apparently US soldiers would use the stock answer "mom and apple pie" when asked why they were enlisting, and/or what they missed about home. I couldn't find a reference for this though. It's not clear why apple pies would be such an important food, especially compared to foods like hot dogs which are closely associated with baseball, except that apple pies would have been baked by mothers and associated with motherhood. However, by 1950 the song The Fiery Bear contains these lyrics:
We love our baseball and apple pie
We love our county fair
We'll keep Old Glory waving high
There's no place here for a bear
Suggesting that by this stage, apple pies were just as patriotically American as baseball, county fairs and the flag.
And in 1975, Chevrolet ran a song with the lyrics "Baseball, hot dogs, apple pie, and Chevrolet", indicating that by then, apple pies were well and truly an American icon.
@congusbongus' answer covered pretty much everything that had to be said about the Apple pies. I wanted to contribute another source to answer your question from phrases.org.uk which adds how American settlers would write back to homeland appreciating American Apple pies and how foreign visitors noted Apple pie as one the first culinary specialities. Quoting my reference below:
"America in So Many Words: Words that have Shaped America" by Allen Metcalf & David K. Barnhart" (Houghton Mifflin Co., Boston, 1997) has a section on the subject --"1697 apple pie."
"Samuel Sewall, distinguished alumnus of Harvard College and citizen of Boston, went on a picnic expedition to Hog Island on October 1, 1697. There he dined on apple pie. He wrote in his diary, 'Had first Butter, Honey, Curds and cream. For Dinner, very good Rost Lamb, Turkey, Fowls, Applepy.'
This is the first, but hardly the last, American mention of a dish whose patriotic symbolism is expressed in a 1984 book by Susan Purdy, 'As Easy as Pie': 'This is IT - what our country and flag are as American as. Since the earliest colonial days, apple pies have been enjoyed in America for breakfast, for an entrée, and for dinner. Colonist wrote home about them and foreign visitors noted apple pie as one of our first culinary specialties.' We cannot claim to have invented the apple pie, just to have perfected it." But here's the surprising part. The expression "as American as apple pie," the authors say, is not that old. "Apple pie figures in our figurative language, too, as in the expressions 'simple as pie' (since everyone supposedly knows how to make apple pie) and, though not an Americanism, 'apple-pie order' . But it was only in the twentieth century, apparently in the 1960s, that we began to be 'as American as apple pie.'"
And of course we remember the notorious H. Rap Brown, whose 15 minutes of fame flickered in the '60s. His immortal quote was "violence is American as cherry pie."
Also these are other few websites with other interesting theories on Americanism of Apple pies:
Throughout the 1700s, Pennsylvania Dutch women pioneered methods of preserving apples -- through the peeling, coring, and drying of the fruit -- and made it possible to prepare apple pie at any time of year. In the vein of many things American, settlers then proceeded to declare the apple pie “uniquely American”, often failing to acknowledge its roots. For instance, in America’s first-known cookbook, American Cookery, published in 1798, multiple recipes for apple pies were included with no indication of their cultural origins.
and from todayifoundout.com there is also an alternate explanation to the phrase and America favouring apples in addition to the above, not quoting it here.
Apple trees and fruit trees and vines in general do not grow well in England because it is too cold and wet and the soils are unfavorable. While there may be a few fruit trees in England, they are in relatively short supply and fruit and its produce has not (since Roman times) been an important part of the British diet, which is best known for "bangers and mashed", boiled beef, crackers and other similar items. Most apple cultivars in England are not English at all, but are French cultivars which is why they have French names like "pippin" and "ruffet". France has a far superior climate for fruit trees and vines.
Just as one evidence of this go to a wine store and look for the "British" wine section. Then compare that to the French wine section.
As a comment on the superiority of the American climate and soil for fruit trees I could cite Peter Williamson's treatise on the geography of New England (1759):
The fruits of Old England come to great perfection here, particularly peaches, which are planted trees; and we have commonly 1200 or 1400 fine peaches on such a tree at one time; nay, of the fruit of one single apple tree in one season, nine barrels of cyder have been made.
When the Americans discovered how well apples grew in America it became the standard practice to plant apple orchards on every plantation due to the marketability of cider, being as good as liquid money. Perry, on the other hand, is not as good tasting here, probably due to dryness and for that reason perry has never been important in America, whereas in England perry and cider were originally equally abundant.
In any case, the practice of growing apples for cider was universal throughout colonial America and this led eventually to the large stock of apple trees which our country still has today and are responsible for the widespread presence and love of apple pie.
I might add that the other answers which claim that apple pie is popular here for 20th-century cultural reasons is completely incorrect. Apple pie is traditional here because of the huge number of apple orchards there are which are a remnant of our colonial past and the dealing in cider at that time.