A product can be marked as purchased either with a tap or with a left to right swipe gesture. Select either of these ways in the application settings.
According to the selected method either tap a product to mark it as bought or swipe it left to right:
When you cross out a product, it gets crossed-out and is shifted down. Every shopping list consists of two parts - the top part, that includes the non-purchased items, and the bottom part, where you can see the bought products.
The product, that you've just marked as purchased, is shifted to the beginning of the bottom part until you close or minimize the app. The other crossed-out products are sorted alphabetically.