This picture from Thomas Calculus 13th Edition Page 981 before the discussion on Green's theorem:
$\mathrm{F}$ is the velocity field. I can't understand how the flow rate through the bottom edge is: $\mathrm{{F}\cdot \hat{i} \Delta x}$ which has units of $m^2 s^{-1}$. It doesn't make physical sense to me. Could anyone explain that?
Say, you have some conserved (or not) quantity with the dimension $\mathrm{Q}$. It can be amount of drug ($\mathrm Q=\mathrm{mol}$), energy ($\mathrm Q=\mathrm J$), it doesn't matter. Then in 2D you can introduce the density $\rho$ of the quantity with $[\rho]=\mathrm Q/\mathrm m^2$. To find the amount of quantity in some region, you need to integrate the density over this region.
Now if you multiply the density by velocity field, you will get a flow $\vec F$: $$[\vec F]=[\rho\vec v]=\frac{\mathrm Q}{\mathrm m^2}\frac{\mathrm m}{\mathrm s} = \frac{\mathrm Q}{\mathrm{m\cdot s}}.$$
Thus, by multiplying $\vec F\cdot \vec i\Delta x$, you will get the dimension of $\mathrm Q/\mathrm s$, which reads “the amount of $\mathrm Q$ that passes through the boundary $\Delta x$ in 1 second”.
Finally, sometimes we are interested in the velocity field itself so we choose entity $\mathrm Q$ to be the unit of scalar density: $\mathrm Q=m^2$, so ($[\rho]=1$).