Why doesn't the curved spacetime curve the stick? We know that gravity is not a force, but a curvature of the spacetime. This is a great visualization.
But I don't understand something. If we live on Earth in a curved spacetime, and this curvature is so significant that an egg rolls off the table at the slightest angle. If we take a long stick which looks perfectly straight in curved spacetime on the surface of the Earth. Then if we take this stick into space away from the massive celestial bodies, it should become curved?
Because in order to appear straight in the curved spacetime, the stick must take the opposite curvature. This opposite curvature in the space will no longer be compensated for by curved spacetime on earth. And the stick should look curved. But we know that this is not happening. Why?
 A: Earth's gravity is quite weak and has essentially a negligible effect on the shape of the stick. Sticks (and basically everything else) are held together by electromagnetic forces and the Pauli exclusion principle, which are the same far from Earth.
Let's say you took the stick to vicinity to a small black hole, so the tidal forces (which are a measure of spacetime curvature) would overcome the strength of the electromagnetic forces holding the stick together. To counter this, maybe you apply some shear forces to the stick to keep in from bending. If you then remove the black hole but keep the shear force applied, the stick will bend. So you are right that gravity can deform an object, relative to its shape in the absence of a gravitational field.
A more "realistic" example of this is a neutron star in a binary. When two neutron stars orbit each other, the gravitational field of one can deform the other. This "tidal deformability" is imprinted in gravitational waveforms and can be used to learn about the internal structure of neutron stars such and the nuclear equation of state.
A: 
If we take a long stick which looks perfectly straight in curved spacetime on the surface of the Earth. Then if we take this stick into space away from the massive celestial bodies, it should become curved?

It certainly can happen that way, but it does not need to happen that way. It all depends on the details of the experiment which you did not specify.
Suppose that we are measuring "perfectly straight" as follows: we have a small tunnel through the axis of the stick with non-reflective walls. We will shine a laser through the tunnel aligned with the entrance of the tunnel, and if it passes through then the stick is straight, if the laser is absorbed we can detect where it is absorbed and know that it is curved in the opposite direction.
Now, suppose that we start with the stick hanging vertically. If it is straight on earth then it will also be straight in space.
Suppose that we start with the stick horizontal supported only on one end (a cantilever beam). In that case a stick that is straight in space will be curved downwards on Earth, and the light will hit the top. Conversely, a stick that is straight on Earth will be curved in space, and the light will hit the "bottom". There is material stress in this stick so the amount of bending depends on the stiffness of the material.
Suppose that we start with the stick horizontal supported on each end. In that case a stick that is straight in space will be curved upwards on Earth, and the light will hit the bottom. Conversely, a stick that is straight on Earth will be curved in space, and the light will hit the "top". There is material stress in this stick so the amount of bending depends on the stiffness of the material.
Suppose that we start with the stick horizontal and supported uniformly across the length of the stick so that there is no material stress in the stick. In that case a stick that is straight in space will be curved upwards on Earth, and a stick that is straight on Earth will be curved "downwards" in space. Since there is no material stress the amount of bending does not depend on the stiffness of the material.
Finally, suppose that we are orienting the stick horizontally and dropping it so that it is in free fall near the surface of the Earth. Further, suppose that the stick is long enough so that tidal effects (spacetime curvature) are non-negligible. In that case, a stick that is straight in space will be curved near the earth and the light will hit the bottom. There is stress in the rod falling near the Earth due to tidal effects so the bending depends on the stiffness of the material.
A: There are a couple of things going on here.
First, the curvature is much less than you think. It takes perhaps a tenth of a second for gravity to have a noticeable effect on the egg. Over that time, the egg's worldline traces out a path with a length of a tenth of a light second, or about 30,000 km. It's only at that scale of thousands of km that the curvature is large enough to be noticeable. The stick's worldline is also that long, but the stick's length (the "width" of its worldline, in effect) is much smaller, so you shouldn't expect to see curvature effects in that direction.
In the diagrams that you linked, the effect of curvature appears large only because the vertical and horizontal axes are drawn to vastly different scales. They're drawn so that 1 second in the horizontal direction equals 10 m in the vertical direction, but 1 s should actually be 299792458 m. If you shrank the diagrams by a factor of 30 million vertically, or expanded them by that factor horizontally, all of the curves would look straight and only careful large-scale measurements would establish that they aren't.
The second thing is that spatial curvature, even when it's noticeable, isn't in a particular direction. If you imagine making a flat sheet of paper cover the surface of a sphere, the paper will crinkle. If you use a more rigid, but still compressible object, it will compress and be under significant pressure if you force it to conform to a sphere. That pressure is the only effect you can actually detect in curved space, because all of your experiments are also confined to the sphere's surface. You can't see the object bending in the background flat space, because the background space doesn't actually exist, it's just a visualization aid.
Sticks are much longer in one direction than the others, and that means there will be little observable effect of spatial curvature even at large scales (a thin strip of paper won't crinkle much when you put it on the sphere). You need an object that is large in at least two spatial dimensions to detect this effect. It also probably needs to be kilometers long in both directions. Even then, I think it would be extremely hard to distinguish the effect of spatial curvature from other effects, such as the ordinary tidal force which is due to its worldline's much larger extent in the time direction.
A: The warping of spacetime absolutely will bend a stick. It's just that most sticks you think of are too stiff to bend.  Take a look at this Willow tree with every branch bent by gravity.  I don't have a photo of a Willow tree in space, but I imagine it would look something like this hair. So what was curved on earth is straightened in space, which is essentially what you are asking. All due to the warping of spacetime.


