I play on Xbox One so idk if that matters but I watched alot of speedruns where people skip to cutscenes or skip past them entirely (also scripted events). How is this done exactly?

Lots of cutscenes can be skipped by pressing a button on the controller, I think Y by default. some cutscenes are not skippable such as the end of Library, or the middle and end of the level Keyes. Some cutscenes are skipped by shooting or being shot at. For example, extending the lightbridge on the level Halo can be skipped by either being shot at by the elites, or throwing a grenade or shooting yourself. The game checks to see if the player is in combat, and certain cutscenes such as the lightbridge, or "shafted" on Silent Cartographer are not played if the player is in combat.