So I heard it from a character in a movie, and looked it up.
If it's correct, why add '-s' to say when the subject is "you"?
As ColinFine says, it is not standard or correct grammar. It's slang.
It means, "So you say" or "That is what you say". The implication is that the statement in question is not true, and the only proof that the person is offering is his own assertion that it is true.
You Ruritanians are all stupid!
The phrase is very informal. It is commonly used in verbal arguments. You would not use it in a scholarly paper -- except to add a humorous touch.
I believe "says you?" arises from the fact that "who" rhymes with "you".
Consider as an example the following exchange between fans of rival baseball teams:
Joe: The Cardinals are indisputably the best baseball team.
Frank: Says who?
Joe: Says me!
Frank: Says you!!!
Your phrase is not strictly grammatically correct, but it flows as a rhyme from other phrases that are grammatically correct and has now become idiomatically correct.
"Says you" is a specific case of a more general way of refusing to agree. When a listener disagrees with a proposition, he may amend it by attributing it to a source. For instance, if you say, "It is good to buy hats", I might respond, "Says the haberdasher" (someone who makes and sells hats). This indicates that I don't necessarily agree that it is good to buy hats, but I agree the haberdasher says that it is good to buy hats.
Because this is an unusual word order (the verb "says" precedes the subject "the haberdasher") it's not uncommon for people to conjugate the verb improperly. As you noted in your question, "you" would ordinarily imply "say" rather than "says".
"Says you" is indeed mostly used to disagree.
But it is also employed as a signal that the preceding comment is unsupported scientifically -- a specific refutation, not a mere disagreement.
This usage follows this idea: If any assertion can be entirely refuted by responding "Says you," then -- even if it's a fact claim -- it is not one that is based on demonstrable evidence.
Refutation by "Says you" is a diagnostic tool to uncover hidden opinions and relative comparisons masquerading as facts ("I own a big dog," "This is how to solve math problems like this," or "You should at least finish out the semester.")
"Says you" doesn't necessarily refute all assertions completely. Sometimes the response can be "No, not 'Says me' -- says 'This evidence right here.'"