Why do object accelerate towards the Earth in general relativity? In general relativity something in free fall, that appear to accelerate towards the earth, is actually not accelerating at all but moving along a geodesic so why does it appear that it is accelerating to us relative to the earth?
Is it because the earths surface is accelerating up into it?
 A: The object accelerates downward as measured by the coordinates expressing time and height that we use. So, in a certain pratical way, the acceleration is real, because we measure it.
One analogy is to associate time to longitude and height as latitude in a region very close to the North pole, let's say 1 km around it. Being at rest at the earth surface is equivalent to travel along a parallel (so keeping the same latitude). But travelling in a straight line, making a chord between 2 points of this circle of 1 km radius, requires going to higher latitudes and later on return to the initial one. If the traveller following the straight line relies on the coordinates, the ratio $$\frac{\Delta Lat.}{\Delta Long.}$$ is not constant, so the movement is 'accelerated'. It can be compared to a stone that we throw upwards. It also goes up until reach a maximum point, and comes back afterward.
We can correct it in the analogy by making a rectangle, using the chord as an edge, and deploying cartesian coordinates for time and space instead of longitudes and latitudes. In this case all works fine, and straight lines are represented as constant ratios between coordinates.
In the real world, it is like being in the stone frame. All other objects in free fall will be travelling at constant speeds for that frame. But it works for small time intervals and small height.
A: 
Is it because the earths surface is accelerating up into it?

It's more like it's just different directions in spacetime. The shortest path between today and tomorrow is through the center of the earth.
In 4-space everything, even objects "at rest" are moving. You can see this when you look at your watch. Sit still all you want, it keeps ticking. The velocity through 4-space is the speed of light, or if you prefer, one second per second. The key to this is that that velocity is invariant: the speed of light is the same in all frames.
So the situation is much like a ball at the top of a hill in terms of conservation of energy. As the ball rolls down the hill, you lose potential energy and it turns into kinetic. In this case, as you "roll down the gravity well" you are losing speed along the time axis (which is being "bent" from the earth's perspectiv) which means you have to gain it in the other axes in order to conserve the original velocity.
A: 
Is it because the earths surface is accelerating up into it?

I am not a physicist. I don't know whether it is proper to say that the Earth's surface is "accelerating up," but when you stand on the Earth, the force that you feel pressing against your feet is due to the Earth's surface preventing you from following a geodesic (i.e., preventing you from freely falling.)
A: The answer to your question is the four velocity and the fact that we happen to live in a universe where the four velocity vector's magnitude has to stay constant.

Even objects "at rest" (in a given reference frame) are actually moving through spacetime, because spacetime is not just space, but also time: apple is "getting older" - moving through time. The "velocity" through spacetime is called a four-velocity and it is always equal to the speed of light. Spacetime in gravitation field is curved, so the time axis (in simple terms) is no longer orthogonal to the space axes. The apple moving first only in the time direction (i.e. at rest in space) starts accelerating in space thanks to the curvature (the "mixing" of the space and time axes) - the velocity in time becomes velocity in space. The acceleration happens because the time flows slower when the gravitational potential is decreasing. Apple is moving deeper into the graviational field, thus its velocity in the "time direction" is changing (as time gets slower and slower). The four-velocity is conserved (always equal to the speed of light), so the object must accelerate in space. This acceleration has the direction of decreasing gravitational gradient.
Edit - based on the comments I decided to clarify what the four-velocity is:
4-velocity is a four-vector, i.e. a vector with 4 components. The first component is the "speed through time" (how much of the coordinate time elapses per 1 unit of proper time). The remaining 3 components are the classical velocity vector (speed in the 3 spatial directions).
$$ U=\left(c\frac{dt}{d\tau},\frac{dx}{d\tau},\frac{dy}{d\tau},\frac{dz}{d\tau}\right) $$\

If in your example, you put a object initially at rest (relative to Earth) into Earth's gravitational field, then General Relativity tells us that the Earth's gravitational field will have an effect on the object, it will slow it down in the temporal dimension (GR time dilation). Now this means that the object's four velocity's temporal component will change. Remember, the magnitude has to stay constant, so the spatial components will have to compensate (change), meaning, that the object will start moving towards the center of mass of the Earth.
You could say that the objects are trying to reach a balance between the velocities in the different dimensions, but rather, one of the ultimate messages of General Relativity is that an objects velocity is not independent in the different dimensions, the object's velocity in the temporal dimension affects the velocity in the spatial dimensions and vica versa.
