The Story of No Man's Sky is in several layers.
Firstly, we have the history of the galaxy - the wars between the various races & how everyone ended up living in space. You will learn much about this throughout the story missions, as well as through conversation with NPCs & interacting with various buildings.

Secondly, we have the many sub-stories and deep lore that you can find through secondary missions, NPCs, buildings, e.t.c

Thirdly, we have the story of what happened to Nada, Polo & the other travellers; how they came to be on the Space Anomaly.

Fourthly, we have the various things that are hinted at repeatedly; but barely explained - such as the world of glass.

Finally, we have the overarching story of what is really going on in the universe, a series of shocking truths that you will discover throughout the Artemis Path.

The lore can be pieced together from various sources, including: NPC conversations; Plaques & Monoliths; Crashed Ships & Freighters; Abandoned Buildings; Boundary Failures; Rogue Data (Remembrance Terminals), as well as the story & secondary missions - The Atlas Path; The Artemis Path; Dreams Of The Deep; and Starbirth.

[Details of these various layers of story, will be added at a later date, in summary & full forms, with full spoiler warnings]

[For now - watch the below video.

If you can ignore the bad voice, the fact that he mispronounces Hirk & that he totally glosses over the enslavement of the Korvax & how they poisoned the Gek first spawn to secure their independence; then this video is a pretty good explanation of a large percentage of the lore of the game.]