Why does the gift widget show even when the customer doesn't have the trigger product? (Essential Free Gift BOGO BXGY app)

This is how the widget is designed. Before the customer qualifies, it shows the Gift locked message, for example "Add 1 qualifying product to unlock a free gift!". This tells customers about the offer so they can qualify for it.

You can control where it appears:

• Product page: on the Product page tab, change The product page widget will be visible on from All products to specific products or collections, such as only your trigger products.
• Floating gift and Announcement bar: choose the pages under will be visible on, or use Exclude specific pages.
• Cart: the cart widget shows in every cart while it's published. There is no setting to show it only after the trigger product is added.

If you don't want a cart message at all, unpublish the cart widget. Offers that add the gift automatically keep working without a widget: the gift is still added when the conditions are met. See Widget Placements and Design.

You can also change the Gift locked text on each placement's Content section.