Understanding Directional Drilling
It is important to note that this is a very simplistic explanation of an incredibly complex operation. The engineers, geologists, and tool pusher know exactly where they are going and how to get there.
When the vertical hole is done, they pull the drill string out of the hole and put a stearable sub and mud motor between the end of the drill pipe and the drill bit. The stearable sub allows the driller to point the drill bit at a slight angle from verticle. The mud motor (hard to describe) actually provides rotation to the drill bit powered by the force of the drill mud being pumped through it.
The driller will run the drill string back in the hole, points the bit at the correct angle and begins pumping drilling mud down the drill string. Now, the drill string cannot be rotated anymore, it must remain still, except for downward pressure. The mud motor rotates the bit and they begin drilling again. Drill pipe will bend, but not very much or you will get stuck. So the angle of deflection has to be small and over a long distance. But eventually the drill hole will be horizonal.