A full understanding of the velocity profile requires an understanding of
calculus. The law for the velocity can be derived as a solution to
a differential equation. One way to do this is to use an equation known
as the Navier-Stokes equation, simplified to handle our case.
An alternative method is to derive a differential equation using Newton's
second law. If you have some background in calculus, you may want to look
at these outside sources:

A consequence of the velocity profile law is that the average velocity of
the blood in the blood vessel is exactly half of the maximum (or central)
velocity:

This means that the we get the same amount of blood flowing through a blood
vessel using the actual velocity profile as though we had blood all flowing
at the same average velocity. But for this imaginary blood vessel with
everything moving at the same speed, it is easy to calculate the blood flow.
The rate of flow is the cross-sectional area times the average velocity: