They say that you can't prove it.
I think 'they' are the people that either don't want it, or haven't thought about it that much.
It's basically a simple process of elimination.
First off. What do we know?
We exist. So what's the reason for that? How did that come about?
We are a part of this universe. If we have the universe, then whatever the explanation for us in particular is comes after that, so it's kind of a secondary question. We need the universe before we can get to us, so it's a reasonable place to start.
So how did the universe come about? Either it existed forever, or it began to exist at some point. Those are the two options.
Big Bang Theory, expansion of the universe, entropy - essentially all of the evidence in cosmology says that the universe we live in began to exist at some point around 13.8 billion years ago. So that answers that.
So we now again have two possible options for where the universe came from. Either it came from nothing, or it came from something.
If we think about it as a maths thing, then nothing is equal to 0. Something could be 1, it could be 3, it could be seven billion three thousand and twelve. Something could be anything.
If we have nothing then the options are: 0+0=0 or 0x0=0 or 0-0=0
'Nothing' has no abilities, no aspirations, no traits, no potential. Essentially, nothing will only ever amount to nothing.
Seeing as we have something (a universe), we must have started with something.
Even if it was '1', perhaps that '1' has the potential to reproduce, so 1+1 would be 2, 2+1=3 and so on.
If the '1' wasn't able to reproduce other possibilities are open to it. 1 could possibly divide itself. So instead of a 1, you could have two halves, each less than the whole, but still at least having come from somewhere. So perhaps our planet is one billionth of a percent of that original '1'.
At this stage of the train of thought we still don't know what that '1' was, or even if it was '1', but we at least know it makes sense while '0' doesn't.
Cosmologists and physicists seem to be pretty much in agreement about this. Because they realise the universe must have come from somewhere, some of them are offering up the possibility that our universe is just one of many. So when we talk about a universe, we mean a closed system that begins with a Big Bang, expands, and finally dies a heat death. So if there is one, or several, they exist within a bigger cosmos.
Each universe begins and ends, but the cosmos itself could possibly be something that exists forever (because scientific method can't tell us anything about it).
So let's try and figure out what the cosmos could be.
Big Bang theory tells us that space, time, and matter (the stuff that everything is made of - elements, chemicals, energy, all that) all began to exist at that point several billion years ago. That means it did not exist before then.
So the cosmos simply must be spaceless, timeless, and immaterial - because those things didn't exist.
The cosmos must be eternally existing for the reasons covered in the maths bit above. If there was ever 0, there still would only be 0, but as there is something then something must have always been there. The first thing to exist must have been there without beginning, otherwise there would have been nothing before it. It also makes sense that it has no end, because a thing that has existed for eternity, but had a lifespan, would have had eternity to live and die already well before it got around to creating a universe.
So immediately our 'something' that the universe came from has a few necessary traits: eternal, outside of space, and not made of anything found solely in a universe.
We can add 'power' to that list. This thing somehow caused a spark that made the universe (or each universe) begin expanding and growing into stars and planets and galaxies that began sprouting plants and bees and lions.
So what kind of things do we know about that might fit the description? Things that exist in no material form, that exist eternally, potentially can exist outside of our space, and have creative power.
The only examples I'm aware of that come close are natural laws, numbers, concepts, and ideas.
All of those examples only have three of those traits, but no creative power.
Concepts and ideas are created products of a conscious mind. Laws require a lawgiver.
Essentially we have whittled down the only possibility to an immaterial, eternal, powerful, spaceless, conscious mind.
As a conscious mind you can add 'intelligence' to the list of traits - it would have to be pretty clever to know how to create a universe.
You can also add 'free agent' to it. If this thing was just a machine, it would have churned out our universe an infinity ago and everything would already have been destroyed in the heat death. Machines pump stuff out as long as they have fuel to do it, but things that are capable of choosing do it when they decide to, and this thing seems to have chosen to wait.
So our list of traits for what the 'something' was that created the universe is now: spaceless, eternal, immaterial, powerful, creative, intelligent, and free willed.
If that isn't the most basic description of a god, I don't know what is.
So to quickly sum it up:
1. The universe (space, time, matter) began to exist.
2. All things that begin to exist have to come from somewhere.
3. That somewhere can't be made of the stuff that doesn't exist yet.
4. That somewhere must then be spaceless, timeless, and immaterial.
5. Only laws, numbers, and ideas are known for certain to have those properties.
6. Ideas require a mind and laws require a lawgiver. So if they exist, then a conscious intelligent mind must exist to provide them. A conscious mind could potentially be spaceless, timeless, and immaterial.
7. An intelligent, conscious mind that exists eternally, is immaterial, outside of space, has the power to create universes, and freely chooses to do so, is the most straight forward definition of a god.
Dumb Questions To Ask At This Stage
So if God created the universe, what created God?
Remember that word 'eternal'? That means 'without beginning or end'. Eternal things don't get created.
You're thinking of 'immortal', which is 'immune to death', but could still have a beginning like a vampire, or a false deity...
How does this prove that Jesus is the son of God?
It doesn't... it doesn't even try to. It doesn't intend to. Jesus does not come into the equation here at all. Who mentioned Jesus? I didn't.
So which God is it? Zeus, Thor, Osiris, The Spaghetti Monster?
From this proof? We don't know exactly. All this proves is that there is a God and he/she/it has all these traits.
Although... Zeus, Thor, Osiris, and The Spaghetti Monster don't share these traits, so we can actually rule them out. Best thing to do is read up on a few possible gods and find out which one matches the evidence.