How does a pterygopalatine nerve block work to treat cluster headache?

Possibly vascular. Although the answer is unknown, the vascular dilation components of headaches, especially cluster headaches, are well known. The pterygopalatine ganglia of nerves primarily affects vascular and secretory functions. Congested vessels, runny nose, are some of the functions. Dilation of blood vessels has long been thought to be part of the pain of headaches.
Blocks pathway. A sphenopalatine ganglion block will affect both sympathetic and parasympathetic preganglionic fibers as well as sensory affervent nerves. Blocking these pathways inhibits the transmission of the nerve impulses which are thought to be involved in cluster headaches.