Let $f(x,y)$ be some function. Then you can visualize $\partial f / \partial x$ evaluated at a given $y=y_0$ as the derivative of a new function $g(x) = f(x, y_0)$. In particular, $\partial f / \partial x \rvert_{y=y_0}=dg/dx$.
Geometrically, visualize the graph of $f(x,y)$: some surface. Take a plane that is parallel to the $x$ and $z$ axes and slice the graph with it at a given $y=y_0$. The surface touches the plane in a curve. That curve is your $g(x)$, and taking the partial by $x$ means taking the derivative of that $g(x)$ by $x$.

Taking the partial derivatives in different directions is like rotating the plane and looking at the curve in which the surface and the plane intersect.
Credit to alamo.edu for the picture.