How to interpret Lagrangian function (specifically not Lagrangian multiplier)

So suppose you want to maximize $f(x)$ subject to the constraint $g(x) = 0$. One way to think about this is to maximize $f(x) - K (g(x))^2$, where $K$ is very large, and then see what happens as $K \to \infty$.

So to optimize $f(x) - K (g(x))^2$, we obtain $$ \nabla f(x) - 2K g(x) \cdot \nabla g(x) = 0 .\tag 1$$ If $K$ is very large, then any reasonable solution will satisfy that $2K g(x)$ is reasonably sized. Since $K$ is large, $g(x)$ will be very small, that is, effectively zero.

Then we see that the Lagrange multiplier $\lambda = \lim_{K\to\infty} 2K g(x)$ (remembering that $f$ and $g$ also depend upon $K$). Thus the Lagrange multiplier is the extent to which the system tries to change $g(x)$ from being zero.

So, for example, if you solve an Euler-Lagrange equation to find the equations of motion of a system of particles held together by rigid rods, then the Lagrange multiplier represents the internal tension in the rigid rods required to keep the particles in their fixed relative positions.

P.S. in equation (1), the dot product represents $$ \frac{\partial f}{\partial x_i} - \sum_j 2 K g_j \frac{\partial g_j}{\partial x_i} = 0 .$$ Thus $\lambda$ is a vector with the same dimension as the output of $g$.