Jenny Price's book Flight Maps has a chapter about passenger pigeons, although the actual process is well summed up by the other response. Before the nineteenth century, they were a valuable supplemental food source that localities exploited every few years when a flock appeared close by. During the nineteenth century, however, railroads meant that a professional pigeon-hunting industry could develop and supply the big east-coast cities every year. At the same time, pigeons became a fashionable food at posh restaurants in New York, and thus demand for them increased. This increased demand and constant extraction by what were essentially industrial hunters, combined with habitat destruction, meant that passenger pigeons essentially vanished quite rapidly in the 1880s. For decades after, people expected them to return, or thought that they had gone to the Amazon or Siberia or something. Their legendary abundance made it difficult for people to imagine that they could become extinct.

A big factor in the extinction of passenger pigeons is that they cannot successfully reproduce in small flocks. They need large numbers gathered just to initiate courtship. So all hunters had to do to doom them to extinction was hunt flocks down below this threshold, which wasn't hard since in such large groups they're easy to find and easy to trap with nets.

Conservationists did start to get worried as flocks grew smaller, and some laws were passed limiting how and where they could be hunted in some states, but as agentdcf says, most people weren't convinced there ever could be a problem with such a prolific species, so these laws went mostly unenforced.