A more intuitive way to look at it (which uses slightly less mathematics) is:
Dynamic pressure is the pressure exerted when a moving stream of air is brought to rest by impacting on a surface.
The moving air has Momentum = Mass x Velocity
But how much mass is there?
Let’s consider a single cube of air moving air with forward area = A and length = its velocity initial velocity V.
Mass = density x volume so the mass of our cube = Rho x V x A
If the air is being decelerated at a uniform rate from V to zero by impacting on the surface, then its mean velocity is 1/2V
So its volume is 1/2AV and its mass is ½ Rho A V.
Momentum = Mass x Velocity so the mean momentum = ½ Rho A V x V
Which is ½ Rho A Vsquared.
Dividing this by the Area A, to convert it into a pressure gives ½ Rho Vsquared
There is a flaw in this explanation, but it is good enough for bar room chat!