When a crack has been initiated in a material, further loading of the material causes stress concentrations at the crack tip. This high stress causes the material to break at the tip and the crack grows. As the material is loaded again, another stress concentration occurs and the crack grows again. But why do we need to unload the material in between? Why doesn't the crack just propagate immediately from the initial notch?
When the material "breaks", it's the inter-molecular bonds that are breaking. Once they are able to form a new bond to the next available molecule the crack propagation stops and stress is reduced.
Once the load is removed, the material is deformed slightly from it's previously unloaded state. Adding load again creates a similar stress compared to the first load cycle causing another cycle of fracture.
Look up brittle vs ductile fracture, those sources will explain why some materials fail instantly while others fail after high number of cycles.