You can tell you've encircled an opponent because one of the provinces (generally a coastal one) will show a resource depot , which will show a resource such as metal or energy piling up which the industry within the pocket cannot use (because it lacks another component such as rares, or because the pocket contains more resources than industry). When you fight divisions within the pocket, you will see a "lack of supply" penalty which gets larger and larger. Your own units may get one too if estimated supply efficiency is poor but hopefully theirs builds quicker.
As regards what to do with a pocket, generally I want to eliminate it ASAP so I can get my units into position for the next attack, don't give your enemy time to build/redeploy new units. I'm never so short of manpower i'd just consider starving them out.
However, I don't want to run my units dangerously low on org either, that risks a successful relief attempt/breakout by the enemy.
Occasionally you pocket such a large number of troops, or enemy in highly defensible terrain, that you cannot beat them in a single battle.
What I do is, launch an attack at dawn, then cancel it as soon as it gets dark, and repeat. As their supplies run down, their penalty gets bigger and bigger, and as they are out of supply, your forces should regain org faster than theirs between attacks.