It solves the problem in the sense that God is the word we use to describe how anything is able to exist at all. And it provides the idea that, while everything we know has a cause, there may be something outside of the world as we know it that accounts for everything. It may also be possible that nothing is uncaused and there is nothing outside of the world as we know it. But somehow this just does not “seem” as likely. It is difficult to explain why, but people who see it this way know what it means. As Krauhammer says, we are not in a position to know for sure about whether God exists, but atheism just appears to be the least likely of the available theologies.
There are some items of knowledge that are not discovered by research. For example we don’t know the square root of 2 by research, or what is sadness, or whether it is wrong to steal. These things were not discovered by experiment or scientific method, but we count them as things we “know”. The existence of God seems more like this type of knowledge, rather than something that’s discovered by science such as the boiling point of water or the speed of light.