The events of Tyranny of Dragons are the most obvious, world-altering events to occur in 5th edition to date. The majority of events that took place in in Rage of Demons did so in the Underdark, while the canon events of Princes of the Apocalypse amounted to the destruction of a few villages, a number of earthquakes, and the Dessarin Valley experiencing some very strange weather for a bit less than a year.
If you're running Storm King's Thunder, however, you needn't worry too much about the story found in Tyranny of Dragons. The book actually references Tyranny for you- the events of the latter are given as the reason for
Annam the All-father's decision to break the Ordning.
If you'd like to suggest the impact of these stories, the most straightforward way is to have the party encounter veterans of the recent dragon-related war, or survivors of the rash of attacks on settlements by dragons. It's been more than a year in-game, so much of the damage should be repaired by now, and besides, the primary focus is on the recent upswing in giant activity; you'll probably want to make casual mention of raids by hill giants and stone giants more often than anything else.