You are asking why gravity causes an object to accelerate from standstill. Now I must assume you are asking about an object with rest mass that will accelerate from standstill (because of gravity).
Please look at this in wikipedia:
in short, the magnitude of the four-velocity for any object is always a fixed constant:
‖ U ‖^ 2 = c^ 2
Now the four speed vector of an object has a magnitude of c always. This is how the universe is set up and this is how the four speed vector is set up.
Now if in your case the object (with rest mass) is standing still (in spacial dimensions its speed is 0 relative to the other object that has a gravitational effect on it), it is still moving in the time dimension.
You see after the big bang, there were only photons, a sea of photons, and all of them were moving at speed c in vacuum when measured locally in space. This was the only speed. But photons moved in the time dimension with speed 0, so they were not experiencing time like we (with rest mass) do.
To start experiencing time like we do, you need to gain rest mass. Once a particle gained rest mass (SSB, baryon asymmetry), those particles with rest mass started to move in the spatial dimensions with speeds less then c. Now the universe is set up so, and these particles had a four speed vector, that was set up so that its magnitude needs to be c always. Since these particles had spacial speeds less then c, they needed to start moving in the time dimension to make up for it so that the magnitude of the four speed vector stays c.
Now in your case, we now know that gravitational zones have an effect on objects inside them, so that the object will start moving in the time dimension slower. This means that if an object is inside the gravitational zone, time will pass slower for it relatively (compared to objects outside the gravitational zone).
Since the gravitational zone causes time to pass slower, and in your case the object is at rest, at a standstill (relative to the gravitational center), this means that its spacial speed is 0 (relative to the gravitational center). Because it is inside the gravitational zone, the object will start moving in the time dimension slowly (relatively) and because its four speed vector's magnitude needs to stay c, its spatial speed will have to compensate, to keep the magnitude c. This means that it will start moving in space. You are asking why it accelerates. As the object moves towards the center of gravity, it will move deeper into the gravitational zone, and the object will slow even more in the time dimension. As it slows in the time dimension, since the magnitude of the four vector needs to stay c, its spatial speed needs to compensate and it will speed up even more.
Which direction will it start moving? Towards the center of gravity. You could now ask, ok, but how will that object determine which direction to start moving in space? This is what we call a geodesic, that is in 4D the shortest path, and the object will start freefalling towards the center of gravity, that is where the geodesic leads.