Q: General relativity in layman's terms (doubt) I'm a high school student. I was reading a PhD thesis by Rickard Jonsson about visualising general relativity. Fig. 2.11 explained the motion of an apple thrown up into the sky by saying that it's not magically attracted to the earth, it's just that spacetime is curved and the motion of the apple corresponds to a straight line on curved space-time. This space-time diagram was shown for the apple:

I've highlighted the part of the worldline corresponding to the apple going up in yellow and coming back down in blue. My confusion is: the slope of the blue worldline (ie the velocity of the apple when it's coming down) is not 0 at any point. It's some constant. But I've heard that in general relativity, when you drop an apple, its velocity is 0 and it's the earth that moves up?

A: 
But I've heard that in general relativity, when you drop an apple, it's velocity is 0 and it's the earth that moves up ?

What you have heard is incorrect. Strictly speaking, the Earth and the apple both move and it is their joint centre of mass that stays still (or defines an inertial reference frame in GR terms). This is true even in Newtonian physics. But the mass of the Earth is so much larger than the mass of the apple that its motion is negligible and can be ignored for all practical purposes. So in simple examples like this one, the Earth is assumed to define an inertial frame.
There are other reasons why the Earth is not truly an inertial reference frame (for example, it rotates about its axis, and it revolves around the Sun) but again in simple examples these effects are ignored.

A: So let's say the picture depicts an apple thrown on earth surface.
Those height markings must be moving upwards at accelerating rate, because they are at constant distance from the surface of the earth, which is accelerating upwards.
So we can not really say if the blue line is sloped or not, by comparing the line to the height markings.
If the apple was dropped where the blue line starts, then we can say that at the moment of dropping the velocity of the apple was zero, and the velocities of the height markings were zero, and the velocity of the apple stayed zero, while height markings started moving past the apple at accelerating rate.
If the apple was thrown where the yellow line starts, then we can say that just after the throw the velocity of the apple was zero, and the velocities of the height markings were 20 m/s downwards, and the velocity of the apple stayed zero, while height markings accelerated upwards.

A: Simply put, the motion of any object in spacetime follows what's called a geodesic. A geodesic is the shortest distance on a curved surface. Geodesics can be understood from a mathematical point of view which will allow you to connect the math to the scenario you are describing. The math for general relativity is graduate level but can still be understood intuitively. The geodesic equation is given by $$\frac{d^2 x^{\lambda}}{d \tau^2}+\Gamma^{\lambda}_{\mu\nu}\frac{dx^{\mu}}{d \tau}\frac{dx^{\nu}}{d \tau}=0.$$ For this equation to make sense intuitively let's look at the normal Cartesian (x,y) plane. Since the (x,y) plane is flat, $\Gamma^{\lambda}_{\mu\nu}=0$, which yields $$\frac{d^2 x^{\lambda}}{d \tau^2}=0.$$ Upon solving this equation we get $$x^{\lambda}=c_1 \tau+c_2,$$ but that's just the equation of line $$y=mx+b.$$ So you see the shortest distance between any two points is just a straight line in the (x,y) plane. Of course this changes when introduced to a curved surface. But the main idea is that objects like an apple are simply following the geodesic curve.
Edit: The $\Gamma^{\lambda}_{\mu\nu}$ term is called the Christoffel symbol and it measures the rate of change of basis vectors at each point on the spacetime manifold. In other words, they just represent gravitational forces. In the geodesic equation the repeated index implies a sum over the range of that index called tensor contraction, that is $$\frac{d^2 x^{\lambda}}{d \tau^2}+\Gamma^{\lambda}_{\mu\nu}\frac{dx^{\mu}}{d \tau}\frac{dx^{\nu}}{d \tau}$$ $$=\frac{d^2 x^{\lambda}}{d \tau^2}+\sum_{\mu=0}^3\sum_{\nu=0}^3\Gamma^{\lambda}_{\mu\nu}\frac{dx^{\mu}}{d \tau}\frac{dx^{\nu}}{d \tau}$$ $$=0,$$ this is called the Einstein summation convention. A great resource for learning gr is Brian Greene's YouTube Playlist "Your Daily Equation", which is linked here. Specifically, watch episodes 26 and 27 for GR content. I hope your quest on learning GR goes great, you can do it!
NOTE: Don't get overwhelmed by the math. It will all make sense in due time. Brian Greene's playlist is made in a way such that it can be made accessible to anyone with an interest. Please don't get discouraged from learning GR because of the math, it will all make sense eventually. It is very possible to learn everything with effort and interest which you have considering your looking into a graduate level physics class in high school! Good luck on your relativistic journey!

A: The blue line mark as distance is measuring spacetime and black line is showing distance or events. So moving on spacetime is in inertial frame. But whenever one moves on distance or events separation line is under force or acceleration because number on spacrtime represent angular speed or curvature of space. So different speed is different events.
Now look at worldline of an apple, as eventline or space is curved line, mass curve space and thus worldline (graph of space and time) is passing through centre of earth, draws apple toward it. Though representing spacetime is not novel, it is similarly curved if accelrated motion is represented by space and time graph.
Curvature of spacetime depends upon curvature of space due to massive body. Curvature of spacetime shows angular speed. If an object moves on spacetime with given speed, then it quickly circle on highly curvature path which is near to the body causing curvature. More curvature means more gravity, more time dilation and more angular speed.
To better understand it, spacetime is equivalent to equipotential surface. A charge moving on equipotential surface is free from electric field. But if it moves radially, then it experience force. This doesn't mean equipotential have no force but moving on that or spacetime is similar to inertial motion or flat spacetime. This spacetime is derived from both coordinates of space and time, which determined by distance from centre. One can visualize spacetime as surface of sphere, allow to move om it but in perpendicular direction feels force of gravity.

A: Yes, the apple doesn't move but the Earth moves up to hit it.
When you hold an apple still and let it go without giving it any force, it doesn't move (velocity is 0) just like Newton said in his first law of motion.  What Einstein said later is that the apple does move through time, and when there's a big mass like the Earth nearby, the path that it takes through time (plus space so now we're saying the apple is moving in the time-direction of spacetime) is bent into the spatial dimension toward the mass. That means the apple rides its time geodesic line, which initially had no space component, on a curve in spacetime so it's a little bit moving in space too and that makes it move toward the center of the Earth. (or orbit it in a different case).
But, here's the hard part. All the particles of the Earth are also trying to follow curved paths that would take them closer to the center of the planet. They can't all do that because there isn't room for them, so they  compress against each other downward, and that compression force essentially accelerates the top particles upward. The surface of the Earth is accelerating straight up, relative to the geodesic line that it wants to follow downward. You know this is true because the Earth isn't getting rapidly smaller.
So RELATIVE TO THE LOCAL GRAVITATIONAL GEODESIC path, the apple is not moving because you didn't give it any force when you let it go, but the surface of the Earth is accelerating up toward the apple because it's being pushed up by the compression of the earth below.
I think that's a really cool way to think about apples falling.
