Because God allows it. It is God, through His permission, who is the ultimate determinant of whether we can save ourselves by cleaving to Him or condemn ourselves by rejecting Him.
Just as God allows evil, He also allows good. He created us neither good nor evil, but capable of both. For us to have a meaningful relationship with God, and not a slave-Master one, God gave us everything we need to be able to choose good from evil.
You see, FK, you are not consistent when you say that God lets us only sin. Fredom is going whichever way you choose. We can't take credit for the evil we choose, but not take credit for choosing God. Our fallen nature chooses evil, but God gave us spirit and reason that can overcome our nature. At no time do we choose God carnally, but whern we are capable of knowing God spiritually. Just as once we choose evil we choose a path that guides us to a perilous end, once we turn to God, He will guide us to salvation.
Well, I have "admitted" to Joe recently that I think that since we can choose TO sin, that it follows that we can also choose NOT TO sin. That's something. :) However, since I think that our fallen nature IS sin, it does not follow to me that we can take credit for the good we do, because it cannot come from us. (The sinning part is "normal" for us so we do take "credit" for that.) We cannot overcome our nature with our gifts at birth. Sure, we are all not born as homicidal maniacs so God did give us "something", but I think it takes something more from God in order to be saved, grace that comes from God after we are born.