You really have to look through the grimoire to get a better understanding. Even then there is a lot of questions still to be answered. I can try to explain some though.
The Traveler has been around for a very long time, The Traveler once was at Fundement(hive world), and was at the Fallen home world as well before arriving in are solar system. When it arrived it brought about the Golden age for humanity. When the darkness arrived The Traveler tried to leave as it had been running from the darkness for a long time but Rasputin crippled the Traveler so that we could strand a fight against the darkness. Thus the darkness was pushed back and held off because of this. The Traveler created the ghost at this time to seek out those who could wield the power of the light to fight the darkness. This is were us Guardians come into play. Basically our job is to help the Traveler heal and do our best to keep the darkness away.
The darkness is never really defined, it is referred to as a very powerful entity but no one knows exactly what it is.
We start out by heading to the moon to fight the hive and claim the piece of the Traveler, in the process the exo stranger is watching us. It is believed she either traveled back in time, or is from a different timeline and traveled through more then likely the vex network to find someone who could destroy the vex. After seeing us fighting the hive and that we are succeeding she calls out to us to find her on Venus. This is when she warns us of the evil powers of the black heart.
Basically the Vanilla story is us finding the black heart, finding a way to access it and destroying the black heart. At this time we do not know who the EXO stranger was talking to in that cut scene as well as we do not know the exact reason why she was so worried about the vex and black garden.
The hive and fallen both followed the Traveler to earth, the hive took up resident on the moon as Crota's armies took over it. There is not to much back story on the fallen but to understand more about the hive and in particular Oryx(which I understand is not Vanilla) go and read the book of sorrows or watch a lore video about the book in youtube.
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Thank you, that did explain a lot. I still hope they explain it in-game instead of just in Grimoire though...
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Yeah, me to. Glad this helped.