The structure of the units is directly tied to the equations they come from. We are used to units of speed, hence their appearance of being less mysterious. However quite logical interpretations can be given to the one you mention:
$Nm$ can be units of work. In this case, it really means pushing with a force in Newtons $N$ over a distance in meters $m$. An unmoving force does not imply a transfer of energy, but a force applied during a displacement do imply a transfer of energy which is the work. In other words, an integral of force over distance.
$Nm$ can also be units of torque and, although a bit more perplexing, it means pushing with a force in Newton $N$ with a lever of a certain length in meters $m$. As vector quantities (the length is a directed length) their product is given by the cross product, yielding the torque, but the magnitude of their product is the product of their magnitudes times a unitless angular factor. Hence, the $Nm$. It has the same units as work while still being more of a "force" because the associated angular displacement is unitless.
$Js$ is units for angular momentum. Since a torque has the same units as energy (because of the dimensionless angular displacement) angular momentum Is the cumulative rotational effect of a torque over time. In other words, the integral of the torque over time.
$Js$ is a unit of action. This would be the most esoteric, because action is not a quantity present in daily life. It amounts to the integral of energy over time. An analogy with angular momentum can be made where in action/angle variables of a classical system, the action is the "angular momentum" that keeps the angle variable "spinning". In any case, it is still an integral over time of energy.
Maybe you start seeing the pattern here. Alot of these "product" units are naturally associated to integrals of a quantity over another one. As such, they are related to cumulative quantities or effets over another dimension.