ACT doesn't really "do" anything. It all depends on how you end up using the data it collects. What it really is, is a parser. It will get a record of your actions (and the actions of everyone else in your party/alliance/mobs) and show you useful stuff like overall DPS, Crit Chance, etc. It's up to you to figuring out what it's telling you and how you can improve with that.
Like... Savage is the type of content it's used in the most. But not because it actually helps you do Savage. People are running Savage all the time, so you can reliably get lots of parses of the same fight and compare all of them. During the actual Savage fight, it doesn't really do much, except let you know who's contributing the most DPS and things like that. The reason why it gets used with Savage is because Savage is the current end-game content and it is pushing people to play their class optimally.
I'd honestly say one of the best uses of ACT is seeing how well you do your basic rotation on a dummy. Do a bunch of rotation loops and see what happens when you shift around oGCDs, if certain ones clip, etc. That's how all the "suggested" rotations get found out; a bunch of people who know their class really, really well get out ACT and do rotations on dummies and experiment around with it.
There's two main tools to use ACT with. The first is
FFLogs, which turns your ACT file into a nice graph with a bunch of tools to do analysis with. The second tool is
XIVanalysis, which analyses your FFLog for you and suggest things you need to improve on. Really, ACT just provides the raw data for you to see what buttons you are actually pressing in-game and when. It's the starting point for a whole lot of analysis on how you currently play your class and what are the most obvious changes you should make to get better.
The few classes I would say it directly "helps" when doing savages (or any content) are classes that buff single targets, like AST and Dancer. They can help their team out a lot more if they buff the DPS with the highest overall DPS instead of just someone at random. It's also useful when you are a healer and a lot of people are dead to know what your raise priority should be. Raising the higher DPS jobs is usually a much better idea then raising the ones putting out less DPS... unless it's a Red Mage or SMN. But like... when I play SAM and use ACT... all it does for me is tell me how much damage I'm doing compared to everyone else. And it's usually a nice ego stroker, since I have a decent idea on what an effective SAM rotation looks like.