How did John Locke prove the existence of God in his books?

John Locke didn't think you could prove that God exists, but that you could prove that believing in God was perfectly reasonable for educated men such as himself. This involved demonstrating that religion was perfectly consistent with reason, something that later generations of Enlightenment thinkers would not accept.
As both a Christian and an early Enlightenment thinker, Locke sought to protect and defend religious belief by putting it on a rational foundation. That meant clearing away what he regarded as all the superstition that had accrued to Christianity over the centuries.
In The Reasonableness of Christianity, Locke argued for a minimum religious framework in which differences of worship between the various sects, churches, and denominations could be tolerated. Each individual Christian must accept Christ as his or her savior, but beyond that, a great deal of leeway would be permitted, allowing each (Protestant) church to develop in its own way.
What was reasonable about Locke's ideas on Christianity was that they corresponded to Enlightenment ideals of tolerance, which were based on the application of reason to religion. This was a time when bitter religious conflict was tearing Europe apart, and early Enlightenment thinkers such as Locke thought that if it were possible to base religion upon a rational foundation, then men would be more likely to realize what they had in common with each other—the universal capacity to reason—instead of their often petty denominational differences.