How do we know or show that God exists?

That's a good question and is asked by a lot of people.
One of the most simple and logical ways of demonstrating the existence of God is to simply look around you. The buildings, trees, birds, humans, mountains and the long list - who made all this?! These are some amazing works of an intelligent mind. Just like we know that a painting hanging on the wall did not paint itself or a building being constructed next door is not constructing itself - we know that all that we see around did not and could not have made itself if we were to think logically.
Can a painting paint itself? - No!
Can a universe as vast, complex and creative as this - create itself because of an explosion? - You say!
That is a good way to demonstrate the existence of God. And that's a short answer, but check out the recommended resource for a detailed read on the subject. It's interesting.

You can also look up the works of Dr. William Lane Craig, Dr. Frank Turek, Dr. John Lennox etc for some amazing resources on the existence of God.