MOSFET saturation relation between Vgs and Vds
\$V_{gs}\$ defines the thickness of the channel under the Gate (in a 3D mosfet model).  Think of this channel as a hallway whose width you can increase as you increase \$V_{gs}\$.  Think of people walking through the hallway as electrons ( \$I_{d}\$). Think of yourself as a god who can force people to have to run through this hallway (force = applied \$V_{ds}\$, which creates \$I_{d}\$)

If the hallway is wider (\$V_{gs}\$ is higher), you can fit more running people (electrons) before people start getting stuck in a tight trafficy environment (current saturation).

If you keep the hallway constant (\$V_{gs}\$ is constant), the there is only a finite amount of people (finite \$I_{d}\$) you can force (increase \$V_{ds}\$) before people start getting stuck and jammed up (\$I_{d}\$ saturates). If you try to force more people into the jammed hallway (try to increase \$I_{d}\$ by increasing \$V_{ds}\$), you will barely be able to. It will be a lot harder for you because the doorway is already jam-packed (\$I_{d}\$ = saturated) with people (current).

I like to think in analogies because all of these phenomena are already experienced by us in everyday life. This analogy gives a rough idea of whats happening under the hood.  :) Hope it helps!