The HotBG could either have been a 1) a piece of the Darkness found by the Vex or 2) the corrupted heart of he Traveler. Either way, it affected the Vex and vice versa. The Hive probably came upon that piece after the Collapse, when the Traveler had just recently been crippled. Knowing what it was, they took it to their darkest ritual rooms, attempting to drain the Light of the Traveler