When the father tells his sons about joining the army, "Obey them, but remember your name and don't take nothing from no man." what does he...

"The Tall Men" is a short story that Faulkner wrote to try
to promote the ideas of patriotism and of the importance of rugged individualism.  It is
in this context that we should understand the quote that you
cite.

What Buddy McCallum is telling his sons is that they
should not lose sight of the fact that they are individuals who matter.  America, as a
democracy, is of course built on the idea that all individuals have value.  McCallum is
telling his sons to remember that and to fight for that idea, even in the face of a
government bureaucracy that is sometimes dehumanizing.

By
telling his sons to remember their names, he is telling them to take pride in who they
are.  By telling them to obey, but also to not "take anything" from anyone, he is
telling them to do what they are supposed to do.  He is telling them to be respectful,
but not to simply obey others blindly.

In these ways,
McCallum is making a statement about what he believes an American man should be.  This
is why he gives his sons the advice you cite.