The argument claims that:
1. God if he exists must be omnipotent, supremely good and our ultimate creator.
2. Therefore an existent God (being supremely good and competent) would have created the best possible world (if he created anything).
3. As the world is inconsistent (between ages and people) it cannot all be the best possible world.
4. Therefore as the world is not the best possible world, God cannot exist.
This argument makes the claim that a perfect God would have created the best of all possible worlds, and that this is not the best of all possible worlds, therefore there is no God.
But how do we know that this is not the best of all possible worlds? What would we compare this world to in order to realize that this is not the best of all possible worlds? All of the other planets which we have discovered seem less ideal, in one way or another, than the Earth. Based on solely empirical evidence, it would initially seem like this is the best of all possible worlds.
One may argue that even the Earth could be better, such as a world without disease, or one in which our bodies are designed in a way that doesn't break down with age.
At this point, when speculating about the "best of all possible worlds," we should ask, "best at what?"
They say you can't make a better mousetrap, and this is because a mousetrap does what it does simply and effectively, but it's only good when applied toward one objective: trapping mice. When applied toward a different objective, such as opening cans, then you could certainly find a better tool than a mousetrap.
In other words, a mousetrap is the best of all possible tools (that we know of), but only at a single application.
The world is the same. It was created for a particular purpose (assumed in the argument), and it may be the best of all possible creations for that purpose, but it may seem like a cruel, unjust place if we don't understand the purpose for which it was created.
I'm not gong to go into what that purpose is, because every religion and denomination has its own view, but theism generally agrees that the world was created for a purpose.
So for the argument to clam that this is not the best of all possible worlds is to assume that the world was actually made for a purpose, and that we fully understand what that purpose is. Some denominations may make the claim that we understand that, some don't.
Nevertheless, this argument has no power to disprove the existence of God. At best, it can disprove individual religions, but I doubt even the most zealous atheist has to patience to understand the doctrine of every religion in order to disprove them all, one by one.
The second option is that this argument could be used to support agnosticism because we do not know what the purpose is, therefore we do not know if this is the best of all possible worlds.
Either way, this is not an argument for atheism. In order for it to be an argument for atheism, we would would have to have a certain understanding of the purpose for which the Earth was created. However, if that were the case, then we would know that the universe was created by a God for a purpose.
In other words, even if this was an atheistic argument, it would be self-refuting.