They left out some details. The way the reaction is drawn would be impossible but the mechanism is correct if you go through each step thoroughly.
In the last step, the O radical abstracts an H radical from the carbon atom which will end up bonded to the O. Carbon radicals are more stable than O radicals.
Next, a peroxide radical abstracts an H radical from the OH. We now have an O radical that can form a single bond with the C radical.