It says you can use APs over several turns to increase chance of hitting the target, but I don't quite understand how it works? How do I use it?

I don't remember exactly how it works, or I would give you detailed instructions. I do remember that it's sort of confusing.

Basically you use it on your target. It takes a whole turn of inaction. When your sniper's turn comes up again, he'll have bonus points to use for taking an aimed shot.

You must attack enemy with sniper mode. Then you can add unspent AP to your chance-to-hit, you can adding them more than 1 turn, until chance is equal 98%.
The maximum amount of AP which you can add, is depends of "Sniper fire" skill.

I don't think there's a cap on the chance to hit with sniper mode. Like you say, it's limited by how much AP you can spend based on your sniper skill. A low-skilled sniper might not be able to spend enough points to get 100% accuracy. My high-level snipers were using snipe mode pretty much exclusively and making 100% headshots every other turn.

Yes, high level sniper can get this result, but AFAIR, there's indication cap on 98.
I don't have SS installed, I can't check so may be you are right and cap is 100%.
Anyway, i hope TS get the idea of how sniper mode works.

Weird~ I dont know about the 99 limit thingy but my current game is different.

Sniper got 120 snipe. Click on the snipe. spend the rest of AP that turn. The next turn spent nearly all, with 18AP unspent. I wait until the third turn to shoot and with my full AP I can afford to shoot many times, as that first perfect shot that cost 120 AP to aim only destroy half of that target VP (very high level officer).

Dani can spend a whole lot less AP considering her snipe is way less.

So Snipe level = amount of AP spent on aiming.

So yeah, Snipe and a sniper gun help you get that visible but oh so very far target. Careful Shot doesnt even come close.

My mod list is Die Hard, Difficulty Grade Change, and Skill Watchdog, if it's of any help.