As others said, in their cultures it is a standard question to ask, with the requisite reply from them to your reply, that as their guest they may offer you something.
If you like earning YMMV points, always take them up on a meal offering. I have had some of the best times I ever had in an AMP when joining them for meal. If you know how to use chopsticks, it will surprise and or impress them, and don't be surprised or offended when they never ever let your bowl go empty. Turn away 10 seconds and it will ALWAYS be refilled.
On one occasion I was invited to stay and join my provider after a session, the owner/mamasan and two other girls for dinner, she fills a small bowl of rice, hands it to me then proceeds to feed me food from the main dish, to my amazement nobody at the table reacted, they just ate like it was normal practice.
This is normal in many many cultures. They don't see communal germs as a thing like we do in the West. First time I had a traditional meal in a village in Afghanistan, my guys were completely taken aback and some even turned their noses up at the layout. In that culture, the meal is served communally, you eat from the dishes or transfer to your bowl with either bread as your scoop or with your RIGHT hand (never ever with your LEFT!). The guys who turned up their noses were invited by the host to not come back ever.