The word 'God' can mean pretty much anything, depending on how you use it. If you say clearly how you will use it in a conversation, then it might be possible to say whether in that particular meaning God exists or not.
Unfortunately, some people use the word 'God' a lot without knowing what it actually means to them, and then they expect others to understand them. Worse yet, many people just regurgitate what they heard from others without doing any thinking of their own.
It is always possible to define God as something that obviously exists, so it's best to choose a definition that we can put to use when trying to relate our experiences to others. Maybe the simplest, but still useful definition is to say: "God is everything." Unless you engage in a highly philosophical debate, it is obvious to most people that 'everything' does indeed exist, and why not call the wholeness of everything 'God'.
(Even in these philosophical debates the question is not so much about God, but about what it actually means 'to exist'. God is a broad and flexible concept, but trying to wrap your head around whether you actually exist or not can be mind-shattering. Imagine what it feels like not to exist, I dare you!)
Why does God exist? The world is out there - even if we bang our heads against a wall very hard. In a sense, we create a vision of God the very moment we give at least some meaning to this three-letter combination.