grad f always points in the direction of fastest increase. Further, the directional derivative, the derivative in the direction of unit vector v, is the dot product . From that it follows that if the rate of change in the direction of vector v is 0, which says that v is perpendicular to .
In particular, along the curve f(x,y)= 1 (or any constant) the value of f does not change so its derivative along that curve (in the direction tangent to the curve) is 0. That is, for v tangent to the curve so is perpendicular to the curve.