How does the phrase, "men killed, and died, because they were too embarrassed not to" in the first chapter explicate the theme of shame and guilt?

The line you ask about from "The Things They Carried" indirectly suggests the idea that the fighting of a war is absurd and uncontrolled and chaotic and that war is not an idealistic endeavor.

In the setting of the story, the men are not fighting for ideals.  They're just doing what they're ordered to and trying to survive.  Specifically, the line you ask about suggests that soldiers' pride and machismo will not allow them to do anything that would embarrass or humiliate them.  They fight not for a noble cause, but to avoid humiliation. 

But this is not negative or shallow.  These men are trapped.  They are in an impossible situation and maybe the only thing they can accomplish is to avoid shame and guilt. 

Of course, shame and guilt can result from fighting as well as not fighting--depending on the circumstances. 

I don't know if explicate is the best word, but the line demonstrates the situationally ironic world the soldiers exist in.  Ideals do not motivate them, but only the need to accept themselves and be accepted by others.