His first line of argument is that some evils can serve greater good. Parents rightly allow a dentist to cause pain to their children for the greater good of healthy adult teeth. And God (if he exists) can cause or allow what appear to be evils in the service of a greater good which we cannot necessarily yet perceive. He states that one important greater good is that without evils and suffering, our ability to have and show concern for others is impaired. We cannot be concerned for somebody's welfare and take action to assure it unless that person's welfare is somehow at risk, and if there is no suffering there is no possibility of that happening.
His next line of argument is the "Argument from the Need for Knowledge", that we are endowed with free choices, and without natural evils our ability to make choices would be impaired in that the choices would have no effect. Also, without the chance to observe evil outcomes, we have no knowledge as to the likely good or bad effects of a choice, and no means ever of obtaining such knowledge. Since knowledge is a good thing, this is a greater good that is served by the presence of evil.
He then argues God's right to inflict harm, that parents have a right to do some degree of harm to their children for their own long-term good, and that God, being so much more than parents to us all, similarly has a right to inflict harm for the good of our souls.
He then goes on to address the quantity of evil in the world, the idea that, even given all these arguments, with things such as Hiroshima, the Holocaust, ther Lisbon Earthquake and the Black Death, God might have overdone it a bit. This is his response.
What he seems to be arguing is that evil is in fact good, in that the more evil there is, the more opportunity there is for people to do good in response. This is a torturer's charter. "I'm only doing this to you for the good of your family, so they can show sympathy to you in your injured state once I have finished with you." It excuses all kinds of evil. It is easy enough for people to do evil believing they are doing good without cover from this kind of fatuous philosophy. If the Hiroshima bomb had never been built, nobody would have been burned by it, and if nuclear weapons had never been developed, there would have been no need for people to campaign for nuclear disamament, and could perhaps have turned to other more substantive forms of good work.
Suppose that one less person had been burnt by the Hiroshima atomic bomb. Then there would have been less opportunity for courage and sympathy; one less piece of information about the effects of atomic radiation, less people (relatives of the person burnt) who would have had a strong desire to campaign for nuclear disarmament and against imperialist expansion. Ando so on. Of course, removal of one bad state or the possibility of one bad state will not remove much good, any more than the removal of one grain of sand will make much difference to the fact that you still have a heap of sand. But the removal of one grain of sand will make a bit of a difference, and so will the removal of one bad state.
It also suggests that there is ultimately nothing that can be done to reduce the total amount of evil in the world - that God decides how much evil there should be, and that everything is always for the best in this the best of all possible worlds.
There are two issues here. The first is how nauseatingly grotesque this philosophy is, and how easily it can be turned to the service of evil actions. The second is how completely without evidence all this hypothesising is. It depends on us making the assumption without specific evidence that there is a God, that he is good, that he allows evil for the the benefit of our immortal souls, that the amount of evil is at its optimum value for God's perfectly good purposes, and that the removal of even one small amount of human or natural evil will remove as much or more opportunity for humans to be good in response. Phooey.
But it gets worse. Swinburne's next argument is the "Argument from Hiddenness", the idea that keeping himself largely hidden from human sight is just the sort of thing a perfect God would do, since too obvious a Godly presence inhibits the ability of humans to make a free choice whether or not to worship God, and that allowing evil into the world is a part of the process of keeping God hidden from view. If you take this argument, then God ceases to be any kind of scientific proposition at all, he becomes a mechanism by which any possible phenomenon can be interpreted as being consistent with God's existence. If there is no possible phenomenon that can be shown, even in principle, to be incompatible with God's existence, then you now have an unfalsifiable proposition, and therefore all Swinburne's ideas of probability, flawed though they were from the outset, go out of the window, because probability has no meaning at all in the context of unfalsifiable propositions, and no evidence one way or the other can affect one's thinking on the matter.
But for all that, he grandly concludes that the existence of evil in the world does not form a C-inductive argument against God, and the probability of God's existence is unimpaired. There is no explaining going on here, merely explaining-away.