My understanding is that finesse is used mainly in instances/raids to make mobs and bosses in those instances/raids more challenging. Regular landscape mobs/bosses don't use finesse, or only use a limited amount of it (some of the landscape warbands in Rohan like Kramp and Bugud might use finesse).
What finesse does is boost the ability of a mob/boss to block/parry/evade and resist your attacks or other skills (a boss or mob with finesse might have an increased chance to resist taunts from a tank, which means the tank builds less aggro/threat on that mob/boss, or the mob/boss might have a higher chance to resist debuffs from an LM or Burg as another example).
The finesse on the gear you equip like Hytbold armor and some other items (some jewelry pieces also have finesse) is to balance out the finesse on the mobs/bosses. If your finesse is higher than the mob/boss, their chance to block/parry/evade/resist you will be reduced, so your skills will succeed more often. If their finesse is higher than yours, they will have a higher chance to block/parry/evade/resist you, so your skills will succeed less often.
The obvious question becomes, "How much finesse should I have?". I really don't know to be honest. I'm not a serious raider, so I don't equip gear because it has finesse. I look at the other stats on the gear and treat the finesse stat as something extra that is handy to have, but I don't select gear because of how much finesse it has.
Therina - Hobbit Guard Rongo - Hobbit Warden
Frood - Man Minstrel Garmun - Man Captain
Zorosi - Dwarf Champ Froodaroon - Elf Hunter
Southern Defenders - Arkenstone (formerly Elendilmir)