Dynamic viscosity represents the resistance of fluid to shear forces as you said. This is what lay people think of when they think viscosity. High dynamic viscosity = more resistance to flow. E.g. honey has much higher dynamic viscosity than water, or cold motor oil has higher dynamic viscosity than warm motor oil.
Kinematic viscosity is something totally different. Don't try to think of it as resistance of fluid to flowing, because that's not what it is. Thinking of it as so many $m^2$ of fluid flowing per second is not what it is about.
To understand what it is, you have to understand that there are many different types of fluid flow, and they behave very differently. One of the fundamental distinctions has to do with the ratio of inertia forces to viscous forces, which is called the Reynold's number. Wikipedia has a pretty good page on it. Flows with high reynolds number will behave completely differently than flows with low reynolds number.
The Reynold's number is defined as $Re = \rho s L / \mu$, where $\rho$ is density, $s$ is velocity, L is a length, and $\mu$ is dynamic viscosity. Of these four quantities, 2 are intrinsic to a particular fluid (density and dynamic viscosity) and 2 are more to do with the situation (the length and the velocity). The reynold's number is super important and it comes up all the time in fluid mechanics. Because it comes up so often, and because both density and viscosity are intrinsic to the type of fluid, we simplify the Reynold's number equation to just $Re = s L / \nu$, where $\nu$ is kinematic viscosity. Now it's 3 equations, exactly 1 of which has to do with the type of fluid, and 2 that have to do with the particular situation. This is just for convenience. We could have just kept the original two quantities.
As for the units, the other two quantities in the reynold's number equation are length and velocity. Length * velocity will give you m^2/s. Note that is the same units as kinematic viscosity. So when you calculate $Re = s L / \nu$, you come out with a pure number that has no units. Unitless numbers occur very frequently in fluid mechanics.