It is true that the air on the "top" (suction side) is faster than on the "bottom" (pressure side). The interesting thing is that the air on the suction side is so fast, it overtakes the air on the pressure side (see illustration from Wikipedia:
The key to understand the velocity distribution around an air foil is not to look at it independent from the pressure distribution and curvature of the airfoil.
Let's start by describing the flow from the stagnation point this is the point at the very left in the figure where the free air flow hits the airfoil first (leading edge) and move along the suction side to the end of the air foil (trailing edge).
At the stagnation point we have a very high static pressure (imagine holding your hand out of the car window when driving really fast). Around the stagnation point the static air pressure is a lot higher than the air around it. From there on the flow is first accelerated (to the point of maximum profile thickness) and then decelerated (until it reaches the trailing edge).
The flow around and air foil is governed by a set of equations (i.e. momentum and energy conservations). These are the Navier Stokes Equations. Without going into detail the equations are not linear. This means we end up with a chicken-and-egg problem. It means one cannot really answer your question in a step-by-step way. The answer to your question is actually (which is not really satisfactory): The flow field which develops around the airfoil is the only possible state this equation system allows for (conserving momentum and energy).
Where as this sounds really discouraging there are some patterns which occur.
Whenever stream-lines are diverging the pressure has to be lower than the surrounding air.
Whenever stream-lines are converging the pressure has to be higher than the surrounding air.
Let's look at an example:
The flow which is approaching the cylinder (left to right) is converging in front of the stagnation point. The distance between the streamlines is getting smaller and the flow is moving around the cylinder. This happens because the high static pressure at the stagnation point pushes the air away from the cylinder and thereby compressing the flow.
When the flow has reached the top (or bottom) of the cylinder, the streamlines diverge. The distance between the stream lines increases because they flow towards the centre line. In order for the flow to flow inwards the pressure has to be lower than the surrounding air. This lower pressure sucks the air inwards, you might also say the higher pressure of the surrounding air pushes the air.
Now, let's look at an airfoil again:
The stream lines (across the suction side) converge first and diverge than until they reach the trailing edge. This means the pressure at the upper side of the airfoil has to be lower. The opposite is true for the pressure (lower) side of the air foil. Here, the streamlines converge which means a higher pressure is present.
When summing up all the pressure forces around the airfoil there is a net-upward force.
I know that this all sounds kind of mambo jambo. The reason for it is, that the model which is used in most of aerodynamics is continuum mechanics. So in a way you are asking why is the model behaving like this, where we actually told the model to behave in that way.
Our model of the flow is based on the observation of fluids. It is a macroscopic view of something a lot more complex. In principle we can explain everything based on the kinetic theory of gases but this becomes impractical when one wants to calculate the lift of an airfoil.
However, think of gas as a bunch of air molecules which bounce around. They hit each other or surfaces. The rate with which they hit each other and surfaces is what is responsible for the pressure we use in aerodynamics. This bouncing is extremely high. Now think of the curvature of the suction side of the airfoil. It is curved downwards. When the air is flowing across the airfoils suction side the room the air can fill out gets bigger. This means that the same amount of air-molecules have more space therefore the rate at which they hit will become lower (hence lower pressure), and since they have more room to move the bulk-velocity of all air-molecules will increase in this direction (which results in higher velocities).