You are of course referring to Poe's idea of the "unity of effect" that a story achieves. His idea was that the author should decide what emotional effect he or she wants to create in their work and then use every stratagem to achieve it through, for example, words or phrases that occur and reoccur in a particular work of literature.
Thinking about the meaning of this phrase when applied to "The Things They Carried," we can see that O'Brien obviously wants us to feel immense empathy for the group of soldiers, and in particular for Lieutenant Cross. Again and again the phrase "They carried" is used to emphasise the way that these soldiers are burdened in a number of different ways, both physically, with military equipment, but also mentally and emotionally. Jimmy Cross, for example, carries his photo of Martha that, in some ways, weighs more than anything else. It is his love for Martha that he "blames" on Lavender's death as he feels he was distracted. Note what he says after Lavender's death:
It was very sad, he thought. The things men carried inside. The things men did or felt they had to do.
The text then goes on to suggest how Jimmy Cross does precisely this by deliberately cutting off his emotional ties to Martha and focussing dispassionately on the job. We witness the way that war is dehumanising the soldiers who are supposed to be fighting it through their relationship with the things they carried, and our hearts ache for these young men, some of them still even teenagers, who so much is being expected of.