Does Earth accelerate towards the object or the object accelerates toward the Earth?
If objects move along the curvature of spacetime and crash on earth, could we also say that earth accelerates towards them? Sadly, I don't totally understand general relativity.
There are a few layers to this. I will start with Newtonian physics, which addresses your question about who accelerates, and then switch to general relativity for the answer which matches the tags you put on the question.
The answer of "does the earth accelerate towards the object or does the object accelerate to the earth" is a matter of book-keeping. What frame of reference are you in? Newton's Law of Universal Gravitation, $F=\frac{GMm}{r^2}$, is a law that applies in an inertial frame. In this frame, both objects accelerate towards each other. Their relative acceleration is the summation of these accelerations caused by equal and opposite forces. I highlight "relative" because that's a key word for your question. The acceleration you are interested in talking about is a relative acceleration between the Earth and the object.
That being said, remember that this motion is more complicated. In an inertial frame, the Earth is hurtling around in an orbit around the sun in an ellipse, and the object is hurtling around on a wibbly-wobbily orbit which is the composition of the motion around the sun and the motion around the Earth.
If we view this in an Earth-centric frame, such as Earth Centered Earth Fixed (ECEF), the story is different. In this frame the Earth is not accelerating. It's position is always (0, 0, 0), which means its velocity and accelerations are also zero. In this frame, the object is doing all of the accelerating.
But if I may be exceedingly pedantic, to avoid confusion, you cannot technically apply Newtons Universal Law of Gravity here. This is not an inertial frame. Remember that in the inertial frame that we started with, the Earth was being pulled on by the object, and thus the Earth was accelerating. Since the Earth is accelerating in an inertial frame, the ECEF frame cannot be inertial. Practical physics people will ignore what I just said. The accelerations of the Earth due to the object are so astonishingly slight that, for all intents and purposes, we can treat it as if it were zero, and the Earth was in an inertial frame. I make this distinction because in any textbook, you will see applications of Newton's Universal Gravity in the ECEF frame. It's a reasonable assumption. But if you start to try to come up with thought experiments that push the bounds, you will eventually come up with one that is so tortured that this distinction becomes the resolution of the paradox in your head. In physics, we like to say that this assumption is valid when $m \ll M$. It is valid when the mass of the object, $m$, is much less that the mass of the Earth, $M$. That notation is used when the difference is so substantial that it's reasonable to pretend one of them is either 0 or infinity. How big is "substantial?" Depends on the problem you are solving. If you're measuring with a yardstick, something may be insubstantial but become substantial when you start measuring with a micrometer.
And of course, you can flip this around. You can look at an object-centric frame. In this case, the Earth is the thing doing the accelerating. However, unlike the ECEF case, in this case it is unreasonable to hand-wave away the non-inertial nature of this frame. Most of the Earth's acceleration comes from the fact that the equations of motion in this object-centric frame are quite different from $F=ma$. They have all sorts of other terms like centrifugal terms and Coriolis terms. (two terms that arise in rotating frames that are reasonably easy to stumble through the math for). In this frame, it's not useful to say that the acceleration is due to the forces of gravity. The accelerations are mostly due to your chosen frame of reference (which just happens to be a frame we calculated by applying gravity in an inertial frame and mentally attaching the frame to the object)
Phew! That was a lot, given that everything above was Newtonian, and your question was tagged for GR. However, I have seen many people stumble because they didn't learn all of those steps and tried to apply the equations intuitively in non-intuitive situations. The above was very pedantic, but it captures things that are often hand-waved away.
The GR story is actually the simple one in freefall. The answer is neither. Neither object is accelerating in freefall in GR, so it's moot to ask which one is doing the acceleration. In GR, both the Earth and the object are moving along their world-line, a geodesic in 4-D spacetime, at a constant speed. If I phrase it incorrectly but perhaps more impactfully: both objects are moving in a straight line in 4-D spacetime. That statement is incorrect, because the concept of straight line is a bit non-sensical in a curved spacetime. That's why we use a mathematical term like "geodesic" rather than a simpler term like "a straight line." This is the message Veritasium was trying to get at in his infamous YouTube video on the topic. The argument is that gravity is not a force, per se. Instead, it is the product of the painful reality that our 3-D world does not have any inertial frames... it does not technically follow Netwon's laws. They could be thought of as ficticious forces, just like the centrifugal force that "pulls" you towards the side of your car while aggressively cornering. It's just such a prevalent ficticious force that it took a few hundred years to realize it wasn't a real force!
If the object is not in free-fall, then we do indeed say that the object has an acceleration in all inertial frames. That acceleration is not due to gravity. The acceleration is due to the electrostatic repulsions between the atoms in the surface of the Earth and the atoms at the surface of the object. In GR terms, that is a true force, and causes a true acceleration in the 4-D spacetime sense.
For most scenarios, you can ignore everything I just said after, "The answer of "does the earth accelerate towards the object or does the object accelerate to the earth" is a matter of book-keeping" in the second paragraph. For most real life practical situations, the non-inertial and/or relativistic effects are so astonishingly slight that you'll still get a sufficiently right answer, even if you didn't model them. The effect are typically way smaller than your measurement errors in the first place. However, if you start going fast, like a satellite, you increase the likelihood that these effects start to become meaningful. At this time, you have to step up the fidelity of your calculations, and start admitting that these effects take place.
I ask if objects moving along the curvature of spacetime and crash on earth how could we say that earth accelerates toward the object.
The objects falling towards Earth curve spacetime, too, which affects the Earth in the same way that the Earth’s curvature affects the objects. We don’t actually have an exact solution for what this looks like, general-relativistically, because Einstein’s equations are highly nonlinear and very difficult to solve for systems of medium or high complexity such as this.
In such weak fields, though, general relativity reduces to Newtonian gravity, and it is okay to approximate gravity as Newtonian, in which case the acceleration of a body $\text A$ due to a body $\text B$ is
$$a_\text A=\frac{GM_\text B}{r^2}$$
directed towards the body $\text B$. In this case, you can trivially see that the Earth is affected by other objects, too, but typically not as much as those objects are affected by the Earth.
The question in the title is
Does earth accelerate towards the object or the object accelerates toward the earth?
The answer that general relativity tells us is: neither.
Both the earth and the object follow a geodesic worldline, which is a path through spacetime determined by the spacetime curvature. Somewhere, the spacetime paths of the earth and the object intersect, due to some set of assumed initial conditions, that lead to an ultimate collision of the two bodies.
When we talk about "who accelerates towards who", what we usually really want to ask is which is the dominant object in generating the spacetime curvature, that hence is dominant in governing the path of the other object through spacetime.
If "object" in your question was a rogue black hole then, one would probably tend to say the earth accelerates towards "object". But if "object" is a piano in free fall towards earth, one would prefer saying "object" accelerates towards earth. It becomes more of a semantic issue, albeit again, not totally devoid of content, since it is a concrete question to ask which of the bodies is dominant in determining the spacetime curvature, or technically speaking, the metric.
Does Earth accelerate towards the object or the object accelerates toward the Earth?
In modern physics there are two distinct concepts of acceleration: proper acceleration and coordinate acceleration.
Proper acceleration is the physical acceleration that you feel and it can be directly measured by an accelerometer attached to the object. Accelerometers on the ground show that the ground has a physical proper acceleration of $g$ upwards. Accelerometers on free falling objects show that they have $0$ proper acceleration. It is in this physical sense that the ground accelerates upwards towards a falling object.
Coordinate acceleration is the mathematical acceleration that is the second derivative of the position in some specified coordinate system. Coordinate acceleration is mathematical and can be changed simply by mathematically changing your coordinate system with a substitution of variables. No physical observation can depend on the coordinate acceleration.
You can choose coordinates where the ground accelerates upwards towards to the object, but you can also choose coordinates where the object accelerates down to the ground. Such coordinates are often distinguished by an inertial force like the centrifugal force or the gravitational force.
