Richard Swinburne summarizes his case for God in the final pages of the final chapter of The Existence of God:

…all that my conclusion so far amounts to is that it is something like as probable as not that theism is true, on the evidence so far considered. However, so far in this chapter I have ignored one crucial piece of evidence, the evidence from religious experience. I concluded the last chapter (p.326) with the claim that, unless the probability of theism on other evidence is very low, the testimony of many witnesses to experiences apparently of God suffices to make many of those experiences probably veridical. That is, the evidence of religious experience is in that case sufficient to make theism overall probable. (EOG, 2nd ed., p.341)1. Based on evidence other than religious experience, the existence of God is not very improbable.
2. If based on evidence other than religious experience, the existence of God is not very improbable, then the evidence from religious experience (in combination with other relevant evidence) makes the existence of God more probable than not.
Therefore:
3. The evidence from religious experience (in combination with other relevant evidence) makes the existence of God more probable than not.
Premise (1) is supported by the following claim:4. Based on evidence other than religious experience, the existence of God is approximately as probable as not.
Premise (4) is itself a conclusion argued for in Chapters 6 through 12 in EOG, where Swinburne explains and evaluates what he takes to be the most plausible a posteriori (i.e. empirical) arguments for and against the existence of God.
Swinburne thus breaks down his case for God into two steps, as represented by the following two-premise argument (my summary of his reasoning):

Premise (2) is supported by the reasoning in Chapter 13 of EOG, called “The Argument from Religious Experience”.