Surface tension. Water is sticky. It tries to attach itself to things. Try to pour water SLOWLY or of a glass. It sticks to the glass as it goes over the edge. If you pour faster, the bonds are broken, and the water flows freely out on the glass, or over the damn.

In this situation, we have to imagine that the water began flowing over the damn slowly as the water rose. As it rose higher, it still had a "grip" on the dam. When the stick broke that bond, The inertia of the flowing water was too much for the surface tension to pull in back.

Similar to what makes drips. The drop gets larger and larger, until it can no longer hold itself up.