The object of ‘give’ is ‘our genius, our soul’ but this doesn’t make sense literally: how can we give our genius? One of the themes of the book is human creativity (and the forces which oppose it, which Pressfield personifies as ‘Resistance’) so the object of ‘give’ must be understood to be the products or works of genius, not genius itself.
There is no mention of whom these works of genius are to be given, so we have to deduce the recipient: something like ‘to the world’ or ‘to the public’ must be understood.
So I would paraphrase this sentence as, “Resistance’s target is the epicentre of our being, our genius or soul: this is a unique and priceless gift; we were put on earth to give its products to the world; and no-one else has it except us.”
Considered charitably, we might describe Pressfield’s version of the sentence as a combination of metonymy (whereby ‘genius’ is used with the meaning ‘product of genius’) and pun (whereby ‘genius’ appears once but is used with two different meanings). Considered uncharitably, we might say that the author allowed his figures of speech to run away with him.