| It is indeed vastly improbable that life will spontaneously emerge at a specific moment in a specific location.
1: However, the Universe is both Large (with many many billions of locations) and Ancient (many billions of years in which to happen).
Add that up, and you result with a significant probability that spontaneous generation of life has happened at any time in the past at any location in the universe.
2: The defining attribute of life, is that it absorbs non-life and turns some of it into life.
3: Therefore, once life has happened once, it is likely to continue to exist, and will spread relatively quickly.
4: Because the Universe works on randomness at very small scales, random 'errors' will occur and life will change.
Many (perhaps most) of these changes will be for the worse, and that particular organism will die.
However, some will be better, and so that particular organism will fare better than its peers, and so reproduce faster.
5: So long as the error rate is sufficiently low, every small change for the better will persist in future generations, and continue to be improved on by pure chance.
Please tell us all which part (or parts) of this you do not believe! | 
