Spacetime diagrams and their interpretation

Minkowski diagrams are related to the Lorentz transformations in the same way a graph is related to its funcion. You don't need the graph of $x^2$ to understand its behavior, but it can come in handy. It's useful, for example, to solve paradoxes like the ladder paradox or the twin paradox.

As for your other question, while it's easy to see length contraction on a Minkowski diagram, it's not that easy to measure it, because you can't just grab a ruler and measure lengths: the curves of constant length in facts aren't the good old straight lines of $R^2$, but are indeed hyperboles.