Bird bones are not necessarily lighter overall than for example mammal bones of similar size; they may be almost hollow, but they are also made of significantly denser material, which can make them stronger but heavier per unit bone. This may be one reason why there's no pressure on other land vertebrates to evolve hollow bones; or put differently: thi smay be one reason why bird ancestors found hollow bones advantageous, as they would otherwise be too heavy (this assumes that the denser material evolved before the hollowness).
They characteristic to make bones hollow probably originated in birds' ancestors at some point in the Triassic Period (about 230 million years ago). At this time in earth's history the atmosphere apparently contained much less oxygen than today (dropping below 15% after the Permian/Triassic Mass Extinction; as opposed to today's 21%). The group of dinosaurs that bird belong to, the Sauriscia, evolved at around this time, and all of them had hollow bones to some extent (including the giant sauropods). One theory is that as the air sacs that filled these hollows are a part of the lung system, the extra surface area allowed them to get more oxygen out of the depleted air. This came in handy later when oxygen levels rose and they could afford to use the extra oxygen for the very oxygen-intensive activity of flying.
After having said all that though, beware that these may simply be "Just So" Stories.