I'll have to treat this with the broadest strokes as you need to further define the question for me to be more detailed.
We could say that life strives to exist because the alternative is nothingness, or nonexistence. Life exists because of random genetic mutations, but life suceeds because of free will and choice.
That leads into the free will vs determinism dichotomy. But that's another subject alltogether. Do you believe your life takes it's course through the choices you make, or do you believe it's all controlled by "fate"? When you answer that question you're at last ready to begin.
As to the origins of life, I would have to say that if it was determinism that brought us to existence, then it was free will that kept us in that state. Of course all of this is from the viewpoint of the evolutionary perspective. Someone else will have to present the religious view. that isn't for me to say.
As to why we have will to live- I belief that's a trait inherent to life on this planet, but it may not be so elsewhere in the universe. We do not know that all life in it's beginnings had will to live. Some may have simply given up on life, thus passing on. While others may have had the determination to exist, to live, however many obstacles it may have had to overcome. And it goes beyond mere living. Not only do humans have to will to live, but also the will and the curiosity to improve ourselves and our life. that is why we consistently evolve. If man did not have this will, this tanacity, then we may never have progressed past the stage of homo erectus. We may never have walked on 2 legs, or even discovered fire. There are other species that evolved much more slowly, or not at all, and others that became extinct. Yet we did none of those. What drove us to set ourselves apart from our ape ancestors? The will to live, and improve.
I don't know if that answers the question, but as I said, I need a more defined question to give you a better answer.