Why aren't checked-off items removed automatically, and how do I clean up?

That's on purpose: checked-off items stay struck through at first so you keep the overview while shopping and don't put anything in the cart twice. To remove them, tap the blue „Clean up“ pill at the bottom of the list (it appears as soon as something is checked off) — or shake your iPhone, if the shake gesture is enabled.

If you'd prefer it automatic, you can set this in an open list via „⋯“ at the top right → „List settings“ → „Customization“ → „Automatic removal of items“. It takes effect when you close the app.

By the way, this does not delete your products: they stay in the product catalog, and you can bring them back with a few taps at any time.