We can show this using $\frac{d\rho}{dt} = 0$, there is no change in the mass density of the flow. But the mass density is a function of space and time. Therefore we can easily obtain the following:

$\frac{d\rho}{dt} = \frac{\partial \rho}{\partial t} + \vec{u} \cdot \nabla \rho = 0$

However, we also have, from the continuity equation.

$\frac{\partial \rho}{\partial t} = - \nabla \cdot (\rho \vec{u}) = - \vec{u} \cdot \nabla \rho - \rho \nabla \cdot \vec{u}$

Therefore, we concludes that a incompressible fliuds, mean that the divergence of the velocity field is zero.

Reference from a question in Physics 105 HW13 Sp24