A Lorentzian metric is the rule spacetime uses to measure the interval between two events, and it behaves differently from an ordinary ruler. Instead of always giving a positive distance, it can come out positive, negative, or zero depending on whether the two events could be connected by a slower-than-light signal, a light signal, or nothing at all.
This is what gives spacetime its built-in sense of cause and effect. Events with a negative interval between them are timelike separated, meaning one could influence the other; events with a positive interval are spacelike separated and can never affect each other, no matter how fast a signal travels.
The Lorentzian metric is the mathematical object that makes concepts like the light cone, proper time, and causal order well defined, rather than just informal descriptions.