Imagine the points of a metric space are whole regions, like US, Spain, and Boston. Distance is "Given the worst case scenario, how far do you have to travel to get from region A to B?"
This actually breaks our symmetry requirement: \(d(Boston,US)=0, d(US,Boston) > 0\)
Which distance is bigger in this framework: \(d(Spain,US)\) or \(d(US,Spain)\)?
\(d(US,Spain)\) is bigger because there is much more room for the worst case scenario to place one farther for Spain.
A bigger first argument makes things strictly worse, all else equal. A bigger second argument makes things strictly better, all else equal.