Ayto doesn't expound how *addrictiāre shifted to mean "direct something, such as a letter, to somebody".
Address originally meant ‘straighten’. William Caxton, for example, here uses it for ‘stand up straight’: ‘The first day that he was washed and bathed he addressed him[self] right up in the basin’ Golden Legend 1483. This gives a clue to its ultimate source, Latin dīrectum ‘straight, direct’. The first two syllables of this seem gradually to have merged together to produce *drictum, which with the addition of the prefix ad- was used to produce the verb *addrictiāre. Of its descendants in modern Romance languages, Italian addirizzare most clearly reveals its source. Old French changed it fairly radically, to adresser, and it was this form which English borrowed. The central current sense of ‘where somebody lives’ developed in the 17th and 18th centuries from the notion of directing something, such as a letter, to somebody.
Word Origins (2005 2e) by John Ayto, p 7 Right column.
Then I researched on Etymonline, but the dis- "apart" baffles me! Firstly, the notion of "apart-ness" doesn't appear in directus. Second, how does the notion of apart-ness relate to the notion of straightness?
early 14c., "to guide, aim, or direct,"
from Old French adrecier "go straight toward; straighten, set right; point, direct" (13c.),
from Vulgar Latin *addirectiare "make straight" (source also of Spanish aderezar, Italian addirizzare),
from ad "to" (see ad-) + *directiare "make straight,"
from Latin directus "straight, direct" past participle of dirigere "set straight,"
from dis- "apart" (see dis-) + regere "to direct, to guide, keep straight" (from PIE root *reg- "move in a straight line"). Compare dress (v.).