First off, I don't get why someone down voted you for just asking a question... So take my upvote.
There is so many things you can do with home assistant therefore not easy to explain all possibilities, here are a few things that comes to mind.
Many smart home users prefer to not depend on cloud giant and that's a way to keep it local and keep everything in just one app even if devices are not same brand. It can just trigger light automatically depending of scenarios (if you have earring issue, it can trigger some light when someone rings at the door). You could have switch that are not well placed and are not practical and it enables you to add wireless switches elsewhere. It can also monitor appliances (and/or services) so it can remind you stuff (useful for some people that easily forget things) like emptying washing machine, taking trash out... (No need to add many daily reminder in your agenda which would have single notification that you will wipe out and forget). You can add security cameras and keep them local only. It can remind you to close windows when you just left home or because it started raining. It is also a way to monitor for exemple water leaks so you can react quicker to avoid big damages (imagine a small leak behind a sink which you wouldn't notice for quite sometimes) same thing applies for smoke detectors, CO2 levels...
The list of things it can do is really endless.
That might depends on how your AC is working, I recently did the same at my dad's house and run into issues because the temp control for the unit is reading from the remote. So when I sent infrared command to turn off unit, it worked as expected for some minutes then turned back on because remote sent back command as temp wasn't meet. Still need to do a lot of testing to find a way to make it work, might need to disable remote and use some temp sensors for the triggering part but not sure how that will work, might need to use some hysterisis friendly climate addon for that...