You should see the Reynolds number as a ratio between the inertial forces and the viscous forces. Higher density of course means higher inertial forces.
So think of a flat plate along which a boundary layer develops (laminar, for now). The outer flow has a certain velocity and the velocity at the wall is zero by definition (the well known no-slip condition, as observed through many experiments). It is the viscosity that inhibits the flow from attaining the free-stream velocity instantly (even with the no-slip condition I mean, this would then cause an infinite gradient, which is of course not physical), so the higher the viscosity the lower the gradient in the flow (and thus the thicker the boundary layer, but I want to keep it to local gradients in the flow).
But, you can flip the argumentation. For equal viscosity, the more momentum is behind the flow, the less viscosity is able to slow down the flow towards the wall, in other words, the higher the momentum, the higher the gradients in the flow.
So, both lower viscosity and higher momentum (either through more fluid, L, higher velocity, V, or higher density, ##\rho##) cause higher gradients in the flow. Turbulence starts when gradients in the flow become too high and thus
instabilities arise. This means that small perturbations in the flow do not damp out and grow towards full fledged chaotic flow (eddies and all that).
So, the short answer: higher density causes higher velocity gradients in the flow, causing flow instabilities, causing turbulence.