If I were to tune my guitar, for instance, from E standard to D standard, why is it better to tune lower than D standard and from there tune up to D standard, versus tuning straight down from E standard to D standard?
The strings are under tension via the machine heads. There is often a bit of slack in the gears. By tuning straight down to D, the slack isn't taken up. By going a bit further down, then coming back up to D, the slack is taken out. Also, the strings are being loosened, and need to go too loose, then tightened up to pitch. Some guitars may not have a problem, but it's worth doing anyway. The old 'mechanism' on violins et al suffer the same, so 'tuning up' rather than tuning down is the order of the day.