What the Source and shards are

The source is our world. The shards are the 13 other versions/dimensions. You are on the 1st shard in SHB.

what the story of the warriors of darkness was all about,

They are our counterparts from the 1st shard, Warriors of Light, and they chose to appear in the early parts of ARR as a Warrior of Darkness. They wanted to upset the balance of our world (Source) to save their world (the 1st). Basically destroy ours to save theirs (more nuanced than that but ELI5).

why I have to go save this other random world now (and from what?) I didn't understand the Garlean Emperor's speech and the true goal of the ascians. I don't get what "a calamity" is, how it's made (sowing chaos?) and why it helps them other than contributing to summoning their God or something.

The 13 shards and source all used to be one singular world/dimension, which was the Ascians world. For some reason it went to shit and they summoned Zodiark to save it, which pissed off another faction who summoned Hydaelyn to stop Zodiark, which she did by kicking him in the nuts so hard it split the one dimension, and thus Zodiark, into 14 parts. Ascians are now working to rejoin these shards together to bring him back, which is done by violently upseting the balance of elements. If a dimension is unbalanced enough it is destroyed/overwhelmed and then rejoins the Source (our world).

Why is it bad? Whenever a rejoining happens the Source is wrecked by mass destruction. One shard being consumed by Wind element and rejoining ours caused the Source (our world) to be wrecked by mass tornados, thunderstorms and hurricanes. It's bad. These are known in history as Calamities.

I don't get why Elidibus / Zenos tried to kill me to "save the star". Aren't the ascians trying to destroy the star?

Zenos doesn't give two shits. Elidibus was 'saving the star' by trying to unbalance it in favor of Light element by summoning armies of Warriors of Light from across the Shards.