There are various ways they might put it, but it boils down to something like: For any model of the universe that includes God, a simpler one can be made for a naturalistic universe.
The way they defend it is by suggesting that there is no reason to have God somewhere in the theory of everything. Everything they study and discover in science uses natural processes, so there's no reason to assume that there is a god at any level of reality. More often than not, the atheist who is making this claim isn't actually a scientist themselves, so any discoveries they have personally found will be minimal and probably previously well documented by actual professionals.
The main defence? Ockham's Razor.
As it is understood by the general public, Ockham's Razor is "the simplest explanation is usually the correct one"
This tends to be the description used by those who make this kind of claim. Unfortunately, it isn't really the best way of putting it.
Say something goes missing from your house. A simple explanation might be 'an alien did it'. It's straight forward, not complicated, and makes sense.
But the better explanation might be something like 'Bob next door took it, he does have a spare key'.
So we have Bob with the spare key, or aliens. What's simpler? Bob and the spare key are two things, an alien is just one. The way the atheist is using Ockham's Razor would suggest that we assume it was an alien, because the job can be done with less things.
Obviously, any rational person is more likely to believe the more complicated theory that requires two things: Bob and a spare key. So 'simpler' doesn't just mean 'less objects'.
What it means is 'less necessary objects', or as William of Ockham originally put it:
"Numquam ponenda est pluralitas sine necessitate"
"Plurality must never be posited without necessity"
So Ockham's Razor is not about the fewest objects, or the easiest to understand theory, it's about the fewest necessary objects.
Look at it this way:
1 + 1 = 2
1 + 1 + 1 = 3
This is simple maths that will help illustrate the point. If you have (1 + 1) you get '2'. If you have any more, then you have something that isn't '2'. So Ockham's Razor is looking to shave away anything more than (1 + 1). In science, it's not always quite that clear, so finding the extra '1's can be a more more tricky.
With that in mind, look at this:
Sometimes in scientific study you see something that you want to explain, and so you come up with a hypothesis or theory based on the evidence you have. You then have to test the theory to see if it holds up.
In the above image, take the diamond to be a conclusion or a fact, and each triangle to be a piece of evidence in a theory.
We can see that you can get the same conclusion from two sets of evidence. But, in the second one, we have more evidence than we need. So that's where Ockham's Razor comes in. We can shave off the third piece of evidence and still have a solid theory that explains the facts. We know that the third triangle has to be shaved away because with it and just one of the other two triangles, the conclusion would be incomplete.
So the question is, is God necessary to the universe, like the first two triangles, or is God an unnecessary third triangle?
When an atheist tries to use Ockham's Razor to prove that God doesn't exist, they are simply making an unfounded assumption that God is a third triangle.
That's clearly circular reasoning. To prove God isn't real, they assume he isn't needed. But, if God does exist, then as creator of everything, he would definitely be one of the first two triangles!
To use Ockham's Razor to shave God away from a theory of everything, the atheist would need to know everything about those first two triangles. That's something far beyond any scientific knowledge we have today.
Imagine a boiling kettle. All the mechanisms are there to make cold water become hot. It all makes sense, but one vital thing is missing. The kettle needs a person to push the button on the side to kick it all off.
So far, when looking at the universe, there are a whole lot of mechanisms that are understood, but there are still a whole bunch that aren't, and when it comes to the question of origins, scientists are generally pretty stuck. They just don't know how a lot of stuff works, and they definitely don't know where it came from. They're working on it, which is great, but they don't have enough info just yet.
In before the cries of "God of the Gaps": there are plenty of philosophical arguments that use scientific evidence to show that God is real. So if we say we don't understand the physical mechanisms fully we don't mean that is the reason to say "God did it". These philosophical arguments resemble scientific theories in many ways and can be put to the test as such.
Any claim that God can be removed from a theory of the universe by using Ockham's Razor is simply grasping at straws that are so far out of human reach, it's ridiculous.