The idea that we should not fight one another is the basic message of this whole poem. So, to find the answer, you simply need to look at the arguments Kirkup makes. The best place to find these arguments is in the part of the poem up to line 12.
Basically, Kirkup is saying that we are all brothers -- that all people are alike. He says that we all walk upon the same earth and we will all be buried in that earth when we die. He says that all people share common goals and hopes and dreams. In these ways, we are all alike. Because we are all alike, it would be perverse to fight one another. It would be as if we were fighting ourselves.
We must not wage war against each other simply because, as referred to in the poem that by harming ones neighbor, which is the main objective of wars, we are, in reality, harming ourselves.