Anri has her hands on a demon blade-- Saika. Being a demon blade, it comes with all the supernatural goodies, and the first one on the list is the ability to control anyone who falls victim to the blade (or in Simple English: if you're cut, Anri is able to control you when she likes).

This is obviously a pretty big thing, so I'll elaborate for you.

Cutting:
Once cut with the sword, the victim becomes Saika's Child. Her (Saika's) Children are connected to Anri (presumably through telepathic means), and this connection allows Anri to control her Children. I'll go through what that means now.

Being Controlled:
Generally, you aren't really aware of the fact that you're handing the reins to Anri. It just happens-- sometimes it'll be her, and other times it's triggered by a perpetual command or one of her personal desires (eg. protect person x at all costs).

When Saika is in control, your conscious - feeling and other mental processes included - is essentially put to sleep. She does what she needs to do, and then you "wake up". You wake up a little dazed (like you would after sleeping normally) after Anri's done the dirty work and released her hold-- but she's generally nice enough to at least make sure you wake up in a way where you aren't totally confused and out of the loop.

That's more or less it, really. If there are any questions you have, then feel free to ask here. This will also function as an opt-in post (not that Anri will ever use Saika to cut and control others unless under special circumstances)!