In everyday life
Like cooking one full plate of a new dish before you cook for 200 guests. Each part tasted fine alone; the plate shows that the sauce and the rice do not go together.
One case
A team adds gift cards to a shop. They build the database for 3 weeks, the server for 2 weeks, then the checkout screen for 2 weeks. In week 7 they connect the layers for the first time, and the server rejects every card code the screen sends. One thin slice in week 1, a single $10 card bought and spent, would have shown the mismatch.
Build one thin piece of a feature through every layer, from the screen to the database, and prove it works. Then build the next piece.
Every case
Each layer can look finished on its own and still fail where it meets the next one. Those meeting points hold most of the surprises: formats, names, missing fields. Work done one layer at a time finds all of them at the end, when every layer is already built on the wrong guess. A thin slice through every layer finds the first surprise in days, while a change of course is still cheap.
The word
Engineers call this
Make the first piece thin and complete: one path, every layer, proven to work. Widen it only after that.