Why is the Gradient of a function f at a point normal to the level set of f at that point?

In the derivation of the technique of Lagrange multipliers, it is stated that where the maximum of f(x,y) occurred, the tangent line of the level set of the objective function f(x,y) and the tangent line of the constraint g(x,y) were the same line. Hence, they had the same normal line. Then, their gradients had to be parallel.

In words, explain why the gradient of a function f at a point is normal to the level set of f at that point.