I really enjoyed the episode. A nice blend a mystery, humour, action and pathos.
So, the Burn was the time almost all the dilithium in the galaxy spontaneously exploded, crippling spacefaring cultures by destroying every warp-capable vessel, along with every other facility that was powered by a matter/antimatter generator.
Fun fact, though - Romulan starships don't use dilithium, they never have. They're powered using contained quantum singularities rather than antimatter, so they don't need dilithium crystals to moderate the matter/antimatter reaction. So, even if they weren't responsible for the disaster, they should have ridden it out relatively unscathed.