With the aid of a high-speed camera, a pair of physicists in France actually figured this one out! For this they won an "Ig Nobel Prize", a parody award given to discoveries that "first make people laugh, and then make them think".
They found that once a piece of spaghetti is bent beyond the breaking point, flexural waves that travel outward through the noodle from the site of the initial fracture. These waves cause the remaining pieces to curve even more before they have had time to straighten themselves out, which leads to additional breaks.
For those of you who want to sound smart, the technical term for the effects of snapping a piece of spaghetti is a 'cascading failure mechanism.'
(Sources: 1, 2)