_Even if it's..._ is more simply . is _just, only_. So overall _Even if it's just the story_.
Since is the object of , it is a bit hard to use _even if it's just.._ here. I'd translate it as _You should listen at least to the story and then conclude_.
Semantically, _at least_ modifies _listen to_. It says, _You should listen to the story at least and conclude, rather than giving the final conclusion without even listening to it_. Also note that the _story_ here means something like _what (someone) has to say_.
* * *
Some other examples:
* Eat miso soup at least! (eating nothing is bad for your health)
* I hope at least you survive ('I' may not be surviving).
* 1 Even if it's just 10000 yen, receiving it makes me happy.