Americans write color and favorite, when others say colour and favourite. How/why did this happen?
The pronunciation is the same, so you can't really say that some "say" this while others "say" that. It's strictly a spelling difference.
These are among the reforms introduced by Noah Webster in his dictionary, with a view towards (a) simplifying the spelling, and (b) creating a distinct American English. (The root forms of many of these words indeed lack the u - for example, Latin color, Italian favorito - so that may have been another motivation of his as well.) So these forms prevailed in the United States, while in the rest of the English-speaking world they kept the original spellings.
The reduction of 'our' to 'or' happens when the ending is unstressed (my accents on the stress):
cólour, flávour, hónour, néighbour, rúmour, lábour, húmour
but not when it is stressed
contóur, velóur, paramóur, troubadóur
This is very well explained (surprise) in Wikipedia
I have heard an alternate explanation - Newspaper reporters would telegraph their stories to the main office for inclusion in the paper. Saving space and reducing the cost of the transmission was important so editors issued a decree to drop 'useless letters' from spelling. Since newspapers were the most distributed mass written product to all levels of American society the spellings they used became the standard.
I have some problems with this - for one I've had a hard time verifying it, for another - I thought telegraph operations were charged by the word - not the letter. From some experience (now 30 years ago - who sends telegraphs now ??) there was a 10 character limit on a word - over 10 characters you got charged for 2 words. Was this the case in the mid 1800's - who knows.
I would image the actual answer is some compound of all of the formal attempts to simplify, common usage and general evolution.
Looking at the graph posted by William, considering that the US-Mexican war started about 1845 and the Treaty of Guadalupe Hidalgo was signed in 1848 (with occupation by US of some Mexican lands, with Mexicans included), and given that "color" and "favorito" are the Spanish spelling for color and favorite, I would say that Spanish had an influence in the current American spelling of those words.
These are all good answers, but I think I have a more likely one.
Scholars in America, the men who influenced politics and academics in the US after the American Revolution were heavily influenced themselves by Latin and Greek sources: Thomas Jefferson and John Adams, for example, were well versed in both and at universities like Harvard and William and Mary studying classical literature was an important part of the curriculum. Noah Webster was a Yale graduate and would have been very familiar with both the local vernacular and have had some knowledge of writers like Aristotle and Virgil and Aurelius. Webster often took into consideration the source of a word, and so often kept to Latin: honorus, colorus, and favorum became honor, color, and favor. He dropped the more complex lettering of Greek based words in favor of the way the Romans would have written it in part because it would have made it much more easy for the man who couldn't afford university to read, with only a few rare exceptions: the word for a doctor that specializes in the human skeleton is an orthopedist in American English and a doctor that might examine you when you are pregnant is a gynecologist. A wine lover is an oenophile, not an œnophile.
Meanwhile, across the Atlantic, Britain's upper class had fallen in love with French. Peppering your speech with French was a sign of education; During this time the British were heavily into international trade and French was its lingua franca. Paris had a very strong hold on the arts and culture as well to the point that people were dumping older words like "fall" for "autumn" and "napkin" for "serviette", and it was this that Johnson took into account when he wrote the dictionary, plus keeping more of the Greek glyphs that often pop up in scientific or technical words (for the doubters out there, look at Kit Marlowe: had an Oxford University education, but he never in a million years would have used the œ and æ of Greek when all he had was the then 24 letters of the English alphabet. He WAAAAAY predates Johnson and though he worked within a system where anything goes in terms of spelling, he was pretty consistent in using only the letters of the English alphabet to spell words of Greek origin, even if he knew the other way, because only a handful of people would have understood it. Johnson changed that and wrote in a time and place where more people could read, so did not bother.)