Edit: I got this backwards. Comment below me has a link that explains caffeine is a vasoconstrictor that relieves headaches caused by swollen blood vessels.
Caffeine is a vasodilator, meaning it causes your blood vessels to expand. This counteracts vasoconstriction, or the shrinking of blood vessels which is a common cause of headaches.
Headache is often a sign of caffeine withdrawal so drinking caffeine will help with the headache in that circumstance. Caffeine constricts your blood vessels so if you stop drinking it the vessels expand, causing pain. If you suffer through it for long enough the headache will eventually go away on its own
Many headaches caused by your blood vessels expanding and creating pressure in your head. Caffiene make the blood vessels shrink, reduces the headache. Other things that might help, drinking water. But especially if the headache is from caffeine withdrawal, caffeine helps.
Edit: typed headache twice instead of caffiene
Because headache is the most promiment caffeine withdrawal symptom, and people who may not be aware that they are experiencing relief of caffeine withdrawal when coffee makes their headache go away may believe that coffee helps with headaches, and thus recommend it to others, not being aware that caffeine only relieves headaches in caffeine dependent individuals and that caffeine is not a general remedy for headaches.
Caffeine is a vasonstrictor (narrows vessels) so it clamps down on vessels in head that can cause headpain, ie headache. The opposite of a vasoconstrictor is a vasodilator (opens up vessels) such nitroglycerin which causes headaches. Another famous vasodilator, sildenafil, opens up some other important vessels (and also known to cause headaches)
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