It's actually pretty simple.
LHS: Rate of change in a quantity $\bf f$ in a volume $\Omega$ is equal to
(1): the rate of change in quantity $\bf f$ in the volume itself, plus
(2): the net influx of the quantity across the boundaries of $\Omega$
The easiest example is mass. If you have a fluid flowing, and an imaginary box (the control volume), the total change in mass inside the box equals the change in density of the fluid inside (usually zero for liquids or low-speed flows of gases) plus the total of the inflow and outflow of fluid across the boundaries.
In the case of mass which cannot have a source or sink in a 3 dimensional fluid, this means that e.g. if you have density decreasing in a control volume, like due to an expanding gas, any decrease in the amount of mass inside the volume must correspond to a net outflow across the boundaries. The mass has to go somewhere. In an incompressiblec flow of a liquid, say, the density doesn't change, so your term (1) is zero, and the equation will require that any inflow must be balanced by an equal outflow. "What goes in must come out elsewhere."
Other things that can be tracked on this way are: momentum, which gives us the velocity equations; energy, which gives us the energy/heat transfer equation; entropy, which gives us yet another equation, usually used more in compressible or turbulent flows.
Unlike mass, momentum and energy can have sources and sinks in the flow, which is why we need to add terms accounting for forces (viscosity, pressure, gravity) that add or remove momentum, and heat sources (viscous dissipation, external heating, internal heat generation due to chemical reaction) that add or remove energy.
But the full set of Navier Stokes equations are taking 3 conversation laws:
Mass is conserved
Momentum is conserved
Energy is conserved
And then saying "each one of these three in any control volume is the sum of the change in that quantity in the CV, plus the net influx/outflux to the CV."