Start in the 1-variable case. The derivative of a function at a point is a number that can be considered a vector that either points left or right. It is by default perpendicular to the level curve which is a point. That vector, f'(c)$f'(c)$, is combined to form (f'(c),-1)$(f'(c),-1)$ which is a vector perpendicular to the graph of the function y=f(x)$y=f(x)$ at the point(c,f(c)) $(c,f(c))$.
For a function z=g(x,y)$z=g(x,y)$ of two variables and for most points, there is a neighborhood of the point at which the level curve g(c,d)=K$g(c,d)=K$ can be written as the graph of a function y=f(x)$y=f(x)$ --- that is a local thing. So the perpendicular to the tangent of that graph (f'(c),-1)$(f'(c),-1)$ is up to a constant the gradient of g.
Finally, consider the case of a linear function. z=Ax+By$z=Ax+By$. The vector (A,B)$(A,B)$ is perpendicular to the line Constant =Ax +By$\textrm{Constant}\; =\; Ax +By$.
