Intuitive

The Minkowski metric encodes the experimental observation that the speed of light has the same value in all inertial frames of reference.

A metric is a tool that allows us to measure the distance between two points. This means, a metric eats two points and spits out a number. Depending on the geometry of the space the points live in the distance can be different. In this sense, a metric tells us almost everything we need to know about the space where our points live.

The stage of special relativity is Minkowski space and the Minkowski metric is the correct tool that tells us how far two points in Minkowski space are away from each other.

Concrete

\begin{equation} \begin{gathered} g_{\mu\nu}=\left( \begin{matrix} 1 & 0 & 0 & 0\cr 0 & -1 & 0 & 0\cr 0 & 0 & -1 & 0\cr 0 & 0 & 0 & -1 \end{matrix} \right) \end{gathered} \end{equation} Thus $x_0=x^0$, $x_j=-x^j$ for $1\le j\le 3$. For real vectors and tensors lowering and raising indices is made by \begin{equation} A_{\mu}^a=g_{\mu\nu}A^{\nu}_a\\ \quad F_{\mu\nu}^a=g_{\mu\alpha}g_{\nu\beta}F^{\alpha\beta}_a\\ \quad \partial_{\mu}=g_{\mu\nu}\partial^{\nu} \end{equation}

Abstract

The Minkowski metric is the correct mathematical tool that tells us how far two events are away from each other in special relativity.

Why is it interesting?