When does a body initially at rest starts moving? Suppose there is a body lying on a rough surface i.e friction is there. Initially the body is at rest.
Let the frictional force be $f$
And the applied force be $F$
So it is said that if we apply a force and this applied force becomes equal to the frictional force then the body will move with constant velocity. 
How is this possible because when the applied force becomes equal to the frictional force then the net force acting on the body is zero i.e.
$$F=f$$
$$F-f=0$$
The net force acting on it is zero so according to Newton's first law of motion it should remain at rest 
because the net acceleration is zero and initially it is at rest so its final velocity after any amount of time should be zero.
But in the books it is written that it will move with constant velocity. How is this possible.
And if it moves with constant velocity then with how much?
I am totally confused here.
 A: 0 is also a constant velocity.
You are completely correct in your analysis here. And they are also correct in their statement, although it is very general. If all forces balance out, then there is no acceleration, meaning  no change in velocity.
If the velocity was 0, then it stays zero. Of it was, say, 10 m/s, then it stays 10 m/s. The velocity remains unchanged. It is constant regardless of its value including 0.
A: If, initially, a body is at rest, the only way to get it moving is to apply a force that will, at least slightly, exceed the static friction.
After that,  the body will start accelerating, according to the Newton's second law, and it will keep accelerating until the applied force is reduced to the level of the dynamic friction (which is slightly lower than the static friction), from which point the body will start moving with a constant speed.
The level of this constant speed will, naturally, depend on by how much and for how long the applied force was exeeding the friction.
If the applied force is reduced below the level of friction, the body will decelerate and stop.
