Something that can help thinking about Euclidean and Minkowski metrics as a programmer, is thinking of how to write a low-level physics sim, and you have a 4D array of values describing some aspect of your simulated world corresponding to space and time and you want to "fill it in" according to some physical rules.

Without any metric, your 4D array entries are conceptually independent and you can't make rules that depend on a neighbor entry (as you haven't defined what a neighbor is!).

In the Euclidean metric, which certainly is the most intuitive, the entries with indices closer to each other in each dimension are also closer to each other physically.

But in the Minkowski metric, the x=0,t+1 or x=0,t-1 entries are actually not closest to your x=0,t=0 entry for example! Instead, x=1,t=1 , or x=-1,t=1 are closer (indeed, with distance zero). Generalizing to more dimensions this maps out what is called the "lightcone" as it's a cone. So you can in some sense say that all points along the lightcone are "the same" or "on top" of the cone tip, at least as seen from your rules that depend on the distance. This difference in metric is what causes all the time dilations and other artifacts of Special Relativity. Light can be seen as a connection between points in the array that are already the "same" point.

You can go one step further and say that the lightcone originating from each array entry can tilt in 4D differently in each point. This is General Relativity and causes all the effects within, if you also postulate that the tilts originate from mass-energy in the region.