I recently had a conversation with a number of people about left handedness. They all had stories from about 50 years ago and they all knew someone who left handed and was mistreated for it. One told a story about a boy getting hand beaten for writing with his left hand at a catholic school.
These are all anecdotal examples, but I assume those types of things happened to people at least sometimes in the past, whereas they never happen today.
Where did this stigma come from? How did it develop?
It's not exclusive to America, left-handedness has traditionally been stigmatised by the Catholic church. The main link for this being that the word for left in Latin is sinister, which is also where the word for sinister in English comes from.
The left is seen to have an inherently negative connotation due to this link, similar to how the number 4 in Mandarin is often considered unlucky, as it sounds similar to the word for death.
In medieval European society, a woman being left-handed was sometimes interpreted to mean she was a witch, left-handed folks were assumed to be working with the devil, etc, so the phenomenon isn't purely American either, but more just came with the migrants.
Sources: here and here. Not the best sources, but this isn't a largely studied topic.