The statement, drinking caffeine can cause headaches, has many additional factors that can lead to the proof of this. When constantly drinking caffeine, your body gets used to absorbing this stimulant along with the feeling of alertness and being awake. When immediately cutting all caffeine intake, your body begins to have withdrawal reactions to the lack of caffeine effects. Taken from UCLA Health, "Caffeine revs up the central nervous system, it can [...] deliver a boost of energy and help with focus and even mood. Caffeine also narrows blood vessels that surround the brain. That is its link to headache."
Headaches are caused by the inflammation of blood vessels which increase the blood flow around the brain. This blood flow increases pressure around the nerves and creates pain. Caffeine can potentially relieve a headache, narrowing the blood vessels round the brain. According to Mayo Clinic Health System, when the body is accustomed to to this effect, "when consumption is stopped, the blood vessels enlarge. This causes an increase in blood flow around the brain and pressures surrounding nerves. This can then trigger what is known as a caffeine withdrawal headache. These headaches can last for a couple of weeks because it takes the body a while to adjust to not having caffeine in its system."
After further reading into the impacts of caffeine on the brain, it is clear that although caffeine can relieve pain, there are also negative impacts during regular consumption. The claim drinking caffeine can cause headaches is true under certain circumstances, which would be true during consistent consumption.