I've sort of gotten a grasp on the Chain rule with one variable. If you hike up a mountain at 2 feet an hour, and the temperature decreases at 2 degrees per feet, the temperature would be decreasing for you at $2\times 2 = 4$ degrees per hour.
But I'm having a bit more trouble understanding the Chain Rule as applied to multiple variables. Even the case of 2 dimensions
$$z = f(x,y),$$
where $x = g(t)$ and $y = h(t)$, so
$$\frac{dz}{dt} = \frac{\partial z}{dx} \frac{dx}{dt} + \frac{\partial z}{dy} \frac{dy}{dt}.$$
Now, this is easy enough to "calculate" (and figure out what goes where). My teacher taught me a neat tree-based graphical method for figuring out partial derivatives using chain rule. All-in-all, it was rather hand-wavey. However, I'm not sure exactly how this works, intuitively.
Why, intuitively, is the equation above true? Why addition? Why not multiplication, like the other chain rule? Why are some multiplied and some added?