I don't think it's nearly as good as "Understand" or "Exhalation" or "Hell is the Absence of God" (all discussed in OP). Like "Life Cycle of Software Objects", it seems seriously overrated. But "Story of Your Life" is definitely one of the more confusing stories and I had to read it at least twice before I thought I understood what sort of Chiang-style message Chiang was going for.
What I take him as suggesting is not that the aliens really are tapping into quantum woo to see the future (this is what inferior SF authors might do and maybe have in the movie which I haven't seen), any more than light rays see into the future in order to take the shortest path, but doing something more like Watts's _Blindsight_ in trying to explore different kinds of minds: that given the reversibility of the laws of physics and the arrow of time, there is no logical or privileged reason to have a conscious experience of the universe as time unfolding sequentially and computationally according to a program computing 1 planck-second at a time, rather than as an Einsteinian 4D block universe in which the block universe is the optimal path between boundary conditions and everything unfolds as it must and already has. This is similar to points Drescher makes in _Good and Real_: a brain could try to understand the universe by taking the present and extrapolating forward but it also takes the present and extrapolate backward. You can 'remember' the future as easily as the past.
The aliens could be doing this, and it is as valid a way of cognition as anything else. And then the drama comes from the protagonist slowly adopting this way of thinking herself to come to terms with her grief over her daughter's death by adopting a timeless Lagrangian way of thinking and explaining it to the reader who is still trapped in the algorithmic perspective.