The unfolding story of the temporal crisis is told through Episodes; compilations of Away Missions and Space Battles that you can engage to de-escalate the emergent conflicts in the galaxy and uncover the mysteries behind the crisis itself. They can be found on your Galaxy Map.
There are two types of Episodes; ones that tell the story of a Dispute between two or more Factions, and ones that seek to uncover more information about the nature and cause of the ongoing anomalies. In a regular Episode, you need only complete all conflicts in the Episode on basic difficulty, but in a Dispute, you will need to ultimately side with one of the involved Factions to put the situation to rest. Doing so will reward you with a Crew member or Ship associated with the Faction that you chose to support. You may need to garner additional favor with them before you can effectively advocate for them; this can be done by completing Shuttle Missions for them. Finally, once you complete an Episode, the next one will appear on your Galaxy Map.
You may also encounter Distress Calls in certain regions of the Galaxy. These function much like Episodes, but are a collection of side Missions not necessary to progress in the story. Typically these will not contain Disputes.