How do I answer the question "Why am I here?" from an aunt who has just entered nursing home (permanently) due to cognitive decline?

I'd try to adjust to providing her information that she can handle in the moment. If she is not able to understand that she is not able to live alone safely, then, she's not likely able to process it and use that information.

Often, the person thinks they are fine living alone. They don't get why it's a problem that they leave the stove on, eat spoiled food, mistake their medication, don't handle hygiene very well, can't work the remote or phone, etc. No matter how much you explain that those things mean they need more care in a facility, they often don't get it. Or, if they do get it, they forget about it. And in some cases they think you are lying. So, they keep asking. I think they may repeatedly ask why, not due to insistence, but, because they have forgotten that they already asked.

I'd pick something that she can accept that's simple. Like, you're here to get better. Feel better, walk better, therapy, all things to help you get better. To me, it's not lying, because, it's actually true. When my LO got to MC, she did feel better. I think she sensed she was getting care she needed, though, she couldn't articulate it. She'd just smile and tell me, They are nice people. lol

If you want to go into other things that she can't process, won't accept and that will get her upset, you can. It'll just be nonproductive and upset you both. Lots of people argue with people who have dementia all the time. It's an argument you can't win.

I told my dad that he was in a facility to get better and build up his strength.

I would avoid the cognitive issues and tell her that the doctor said she needs to have this care to get stronger. I don't think I would say this is your home now, just you need to stay until the doctor says you are strong enough to go home, but only when she asks.

It’s asking too much of her to expect her to understand her own decline and the reasons for her to require the new level of care. You’ll see people on this site refer to the “therapeutic lie” it’s used when a person is no longer capable of understanding a situation. I’m sorry you’re in this position, but it’s far kinder to make up a plausible story that she’ll accept than to try to make her understand her decline