fluids are a shitload easier to solve when you can assume that they’re incompressible. with that assumption in mind, what you don’t want is fluid velocities that are pushing into each other or away from each other, because this would result in fluid densities increasing in some areas and decreasing in others… i.e. compression.
the non-divergent projection iteratively goes over your velocity field and finds ways to redirect those velocities so that they’re not going to result in an increase in pressure. this redirection manifests itself as swirly behavior, which is typically what you want with fluids.
you could see this pretty quickly in action with a 2D pyro sim… make a little ball of gas and apply a uniform outward force from the center. without the non-divergent projection step, the gas would simply expand. with it, it instead starts to swirl around itself in an effort to not increase or decrease pressure.
i’m probably leaving out details that one of you actual smart people can fill in, but that’s my understanding of the process