In Lagrange multiplier method, e.g. optimize a function $f(x_1, \dots, x_n)$ under a constraint $g(x_1, \dots, x_n) = 0$. There is a fact that $\nabla f$ is parallel to $\nabla g$ which is given rise from the level curves of $f$ and $g$ are tangent to each other (i.e. there tangent lines are parallel, then because gradient and tangent of level curve are orthogonal implies the fact above) at the points when $f$ optimized under constraint $g$.
The only part I don't have intuitive understanding is that why level curves of $f$ and $g$ are tangent to each other at where $f$ optimized under $g$.