I will try to sumarize it the best i can, it's confusing.
Keep in mind that it seems like all the events happen in at least two different, or maybe more, timelines.
1.- Lunaria, who claims to be the embodiment of Altea, appears in our world.
2.- She needs to get to the Monolith, but she is defenseless. She encounters a boy (Assassin) and ask him to escort her to the Monolith.
3.- They bond during this travel and develop romantic feelings for each other.
4.- Lunaria is brough to the place she needs to go, presumably near the Monolith, but she doesn't want to go anymore. It's something she must do because there is no one else, but she wants to stay in this world with Assassin.
5.- Hoping that he understand the indirect, Lunaria gifts Assassin her time traveling abilities expecting him to use them to take her away with him. Without her powers, Lunaria is now unable to control her powers and she will randomly time travel against her will.
6.- She heads ahead, but turns around with a face that screams "you fishing idiot use that I know we both have feelings for each other I swear if you don't it you will regret this forever" (according to Illusion) and Assassin understood the indirect.
7.- Assassin takes Lunaria and leaps through time with her, taking them both away from the Monolith.
8.- Lunaria then regrets everything, because of what I said in point 4. And loops the time hoping there will be a different outcome.
9.- There is no different outcome, it's settled that it's her destiny. But she keeps trying and keeps looping. She eventually grows insecure too and despite encountering Assassin in all timelines, their relationship is more and more tense.
10.- Assassin goes insane for having to live through all those loops. He grows up and enrolls into the Dragon Followers in order to be able to find Lunaria (in our current timeline) under the name "Illusion". He is admitted with the promise of bringing Lunaria to King Feather. Also he still has Lunaria's time traveling powers.
11.- He finds Lunaria, but flees. He is called out for being a traitor and condemned to death penalty. He is chased by minions while he was still with Lunaria and fatally wounded. He uses the last of his strength to travel through time to a place where his younger self is and possesses him, so he could still stalk on Lunaria.
12.- Illusion spends a good chunk of his screentime teasing Assassin, but also warning him about Lunaria being dangerous and that she is just manipulating him. Still Illusion keeps saying that Lunaria will be his, I think guy went full Yandere.
13.- I don't know exactly what happens here, but there is a point when Assassin, Lunaria and Illusion travel to met yet another Assassin and kills him in front of everyone. Possible interpretations:
In point 11, Illusion did not only travel through time, but also through space. So the Assassin possessed in our current game wasn't him, but him for another timeline. And in point 13, he traveled back to his original timeline and killed his younger self.
It's all in the same timeline, and the Assassin killed is someone who is the future self of out current Assassin and the past self of Illusion, someone in the middle. Forcing Lunaria (and Assassin) to finally get her **** together and solve the bull**** they got everyone into.
It was just some other Assassin from another timeline killed.
The problem with the first two theories is that Illusion would stop existing if he kills his younger self, anyway....
14.- At this point even present Assassin is mad at Lunaria and says he will never forgive her, and that he can't let her go and use his other selves the way she pleases. Assassin is giving vibes of finally turning into Illusion, but honestly I can't blame him.
I don't know what happens after this. I know that he encounters Lambert at Anu Arendel and tries to kill him to fulfill Assassin's sentence, but is saved by Lunaria.
Sorry if I made a mistake, I have yet to complete Assassin story myself. I'm just trying to recall everything I have read so far.