I'm no expert yet, but as far as I understand from a mechanical engineering perspective you can think of the cpu and heatsink surfaces as solid and imperfect. When you place two solid and imperfect surfaces together they do not interface perfectly.
This would create small gaps between the surfaces that you could never actually see. The thermal paste acts to fill these gaps, letting heat get transferred more stably between the surfaces, creating a continuous path for heat to get transferred to heatpipe/waterblock.
So that means that the two primary properties of a thermal paste are how well it conducts heat itself, and how well it forms itself into the structure (Geometry vs conductivity)