Why does it take such a small incision for the glass to break at that spot? Why is the structural strength of the material influenced by such a small imperfection?
Suppose you bend a perfect, i.e. unscratched, piece of glass, the forces on it look like:
The top of the glass is in tension and the bottom in compression, but the stress is spread over a large area of glass so the local stress at any point isn't enough to break the glass.
Now put a scratch in the top surface and bend it again:
This time the stress is concentrated at the leading edge of the scratch so you get a high local stress and the glass breaks at the tip of the scratch. This means the crack grows until it breaks through the glass. That's why putting a scratch on the glass causes it to break when bent. The scratch focusses the stress at the scratch.