Thinking about a slightly easier example: f(x, y) would be a function which can be visualized in three dimensions: x, y and f(x,y). You can think of f' as the slope of the plane, tangent to the function at (x, y) . If y is kept constant while x is varied (ie df/dy = 0), f' would be equal to df/dx, and if x is kept constant and y varies (df/dx = 0), f' would be df/dy. If both x and y are free to vary, then it makes sense from a graphical point of view for f' to be linearly related to both df/dx and and df/dy, meaning that f' = df/dx plus df/dy, satisfying the functions we previously got when df/dx = 0 and when df/dy = 0. Extrapolating this to three variables x, y and z: f' = df/dx plus df/dy plus df/dz.
Hope this somewhat helped.
Edit:
Sorry about the "plus"es, my phone doesn't seem to be able to write plus signs for some reason