In that context, "upon" means: just after or as something happened. You could reword the sentence as follows:
On February 1st, 2003, the seven crew members of the Space Shuttle Columbia perished when the shuttle disintegrated as it re-entered Earth's atmosphere.
On February 1st, 2003, the seven crew members of the Space Shuttle Columbia perished when the shuttle disintegrated just after re-entering Earth's atmosphere.
However, the word "upon" is a better fit here, for a variety of reasons explained below.
Why this makes sense to native speakers
The primary sense of "upon" is, as you might expect, "above something and on it", as in "A cloth was spread upon the table" and "A crown sat upon the queen's head." The primary sense is rare in English today, and sounds a little archaic, as "upon" has taken on many secondary senses, and the primary sense isn't really much different than plain ol' "on".
One of those secondary senses occurs in this sentence: "Once you reach the top of the hill, you can look upon all of Los Angeles." Here, the suggestion of being above applies to the person doing the looking: since you are up on top of a hill, you are higher than Los Angeles, so you can see the whole thing in front of you.
This sentence contains the primary sense and suggests that secondary sense without actually mentioning looking:
We stand upon the threshold of a new year.
This is a poetic way of saying that we are now at the boundary between the old year and the new year, and we are in a position to look ahead and see or imagine what the new year might bring. Literally, a threshold is the bottom of a doorway that enters into a room. It forms a common metaphor for being at the end of one time and the beginning of another. Here is a famous couplet:
Leaving the old, both worlds at once they view,
That stand upon the threshold of the new.
--Edmund Waller, 1686
The phrase "stand upon the threshold" strongly suggests that you are about to step forward toward the new state of affairs that you now see ahead of you. There is a natural analogy between moving forward in space and action unfolding in time.
Hopefully now it only requires a small step to understand the temporal sense of "upon" in your example. The new event (the shuttle's disintegration) follows immediately "upon" the triggering event (the re-entry). The triggering event is like a threshold that you step across, because it causes a transition from the old state (when the shuttle was OK) to the new state (the shuttle has fallen apart). Notice that re-entry is itself a transition. "Upon" suggests physical contact, like one's foot on the threshold as one enters the room. In your example, the disintegration happens just as re-entry happens: there is a sort of "temporal contact" between the two events.