What does the idiom "to sound a little off" mean in American English?
It means “to sound not quite right”. Three rather literal examples plucked from the web:
“Sometimes the bass on music will sound a little off, almost distorted.”
“it sounds a little off ’cause i still haven’t tuned it”
“Starting this morning when i turned the truck on i noticed the engine sounds a little off. Kinda like one cylinder isn’t firing. ”
It can also have a slightly metaphorical use, to describe a situation that is not right: e.g., “the price they quoted sounds a little off”.
In a related vein you have “to feel a little off” meaning to feel not quite right, applying both to people and things: “I feel a little off after eating the whole tub of icecream”, “the atmosphere of the meeting felt a little off to me”.
It's from off-key [music] pitched higher or lower than the correct notes of a melody.
The word little is arbitrary - in other contexts you might say "That sounds really off to me".
From which it can be seen that the original/literal sense implies a bad fit with the surrounding context (i.e. - the meaning or style of preceding utterances). But in practice today people usually use it to mean a bad fit with my ideas on what you should be saying, or how you should say it.