Why does Leibniz believe that this is the best of all possible worlds?
Leibniz thought this was the best of all possible worlds because God allowed spirituality and reason to coexist and because God allowed evil to exist. With spirituality and reason coexisting, Leibniz hoped to reconcile traditional religious views with the progression of philosophy and also to get different denominations of Christianity to be in harmony.
Most of all, the existence of evil makes this the best of all possible worlds because it makes us appreciate the good (and Leibniz says that the good largely outweighs the evil in the universe). So this evil is a kind of 'original sin' where the human will has the freedom to choose right from wrong and this choice is part of God's divine 'preestablished harmony.' This is where Leibniz believed he had complemented God's preestablished harmony with human freedom. To sum up, he thought this was the best of all possible worlds because evil exists as a choice, there is freedom in human will to choose between good and evil, evil makes us appreciate the good and therefore choice is inherent in finding meaning, and that rationality (reason) coexists with faith. Rationality (reason) includes this freedom of choice but that will power is ultimately known completely by God because God is infinitely good and omniscient.
In general, Leibniz was trying to harmonize religion with reason: the grandeur of God and spirituality with the fallable, yet incredibly potential goodness of the human will.