To understand the proof of the multivariable chain rule, I think you will find it helpful to look first at an informal, nonrigorous demonstration, as given http://math.ucsd.edu/~wgarner/reference/math10c_su10/lectures/chain_rule.pdf (you need only look at the first page of that document). That demonstration uses the fact that if $z = f(x,y)$, and the variables $x, y$ are altered by small amounts $\Delta x,\,\Delta y$, then the corresponding change in $z$ is given by the approximate formula $\Delta z \approx \frac{\partial z}{\partial x}\Delta x + \frac{\partial z}{\partial y}\Delta y$. What the Swedish proof does is to take that informal approach and make it rigorous, replacing the approximate formula by an exact formula of the form $\Delta z = \frac{\partial z}{\partial x}\Delta x + \frac{\partial z}{\partial y}\Delta y + E(\Delta x, \Delta y)$. In that formula, the error term $E(\Delta x, \Delta y)$ represents the amount needed to convert the approximate formula into an exact equation. The essential fact about $E(\Delta x, \Delta y)$ is that it is small compared with $\Delta x$ and $\Delta y$. This is expressed by writing $E(\Delta x, \Delta y)$ as $E(\Delta x, \Delta y) = \rho(\Delta x, \Delta y)\sqrt{\Delta x^2 + \Delta y^2},$ where $\rho(\Delta x, \Delta y)$ is a function that tends to $0$ as $(\Delta x, \Delta y) \to (0,0).$
To sum up, make sure that you understand the ideas in the nonrigorous argument first, then go back to the rigorous approach and see what you can make of it.
