In particular, what instruments do I begin hearing at the ~39 second mark of this video? https://m.youtube.com/watch?v=EO1FZEUbZJE
There's a full symphony orchestra: strings, woodwinds, brass and percussion. It sounds like there is also a full pipe organ playing. It becomes a bit more prominent around the 39s mark. Maybe that's what you are hearing.
Classical composers often added a full organ at climactic moments.
Richard Strauss adds the organ on the last chord of the opening fanfare of "Also sprach Zarathustra" (around 1:28 here).
Gustav Holst adds the organ at the climax of "Mars" from his suite "The Planets" (around 6:06 here)