According to Einstein's theory of relativity, gravity is not pulling us down but space is pushing us down. This mean there is no gravity and its an illusion. So, in this scenario how does Earth hold its atmosphere? Since there is no force to pull it down.
This mean there is no gravity and its an illusion
This is a very common statement in popular science accounts of general relativity, and while it's true it's also misleading.
If you release an object above the Earth then that object will immediately start accelerating towards the Earth. Isaac Newton would say this is because there is a force pushing the object towards the Earth and that's what we mean by the gravitational force. Albert Einstein would say there is no force pushing the object towards the Earth and the object accelerates towards the Earth due to the curvature of spacetime. It is in this sense that we say gravity isn't really a force.
But in both cases the object accelerates towards the Earth, and if the object is an air molecule then that air molecule accelerates towards the Earth. So the Earth's atmosphere stays with the Earth because the air molecules that make up the atmosphere are all individually accelerating towards the Earth just like any other object does. The fact the atmosphere is bound to the Earth is no more surprising than the fact you and I are also bound to the Earth.
I suppose the surprising thing is why the atmosphere doesn't all fall immediately to the Earth's surface to form a thin dense layer of air molecules. The reason this doesn't happen is that air molecules are all whizzing around at surprisingly high speeds - typically hundreds of metres per second depending on the temperature. The air molecules bash into each other and knock each other around, and the air molecules near the ground bash into the air molecules above them and stop them falling down.
According to Einstein's theory of relativity, gravity is not pulling us down but space is pushing us down.
That's not what General Relativity says.
It says that gravity is due to the curvature of space-time.
This mean there is no gravity and its an illusion.
It means no such thing.
It means that we have a deeper reason for the effect we call gravity than we used to. It does not mean the effect (gravity) has vanished.
So in this scenario how earth holds its atmosphere since there is no force to pull its down.
What holds you down (regardless of how you think of it working) will also hold down the atmosphere.
Note that a planet's atmosphere can be lost over time. You might find this Wikipedia page on Atmosphere Escape interesting. So gravity is not necessarily enough to keep an atmosphere bound to a planet.