Two things: a) you didn't quote the full sentence; b) about 99.9% of sentences out there aren't really difficult if you know how to read them.
Here is the sentence in full:
The Universal logo changes to show ships leaving 25th Century Earth as a teacher's voice-over describes the exodus from "Earth that was" and the terraforming and colonization of planets in the new 'verse.'
Let's bracket the sentence to make its parts clearer:
[ The Universal logo ] changes to show [ ships ( leaving [ 25th Century Earth ] ) ] as [ a teacher's voice-over ] describes [ [ the exodus from "Earth that was" ] and [ the terraforming and colonization of planets in the new 'verse.' ] ]
Basically, your sentence is:
SOMETHING1 changes to show SOMETHING2 as "a teacher's voice-over" describes SOMETHING3 and SOMETHING4.
Let's take a closer look at the sentence now.
(That SOMETHING1 is "The Universal logo".)
Changes? For what purpose?
"It changes to show SOMETHING2."
(That SOMETHING2 refers to "ships (that are) leaving 25th Century Earth".)
What happens during the changes?
"It changes while "a teacher's voice-over" describes SOMETHING3 and SOMETHING4."
(A teacher's voice-over = the voice-over of or by a teacher)
(SOMETHING3 = the exodus from "Earth that was"; "Earth that was" is just a name of a planet1.)
(SOMETHING4 = the terraforming and colonizing of planets in the new 'verse' (a new "universe").)
1"Earth that was" possibly refers to our earth, but I haven't seen Firefly so I'm not really sure.
Edit note: TRomano's comment urged me to look for the so-called "pre-trailer" on YouTube. (Because I thought this "as" is more like "because" at first even though I could mean "while" as well; I chose to read it as "because" in my first revision, which was wrong. The clip makes it clear that the sentence is just a description of what happens in the trailer (as opposed to why the trailer was made that way).)
Here is the link to the trailer I found on YouTube: https://www.youtube.com/watch?v=WTVdh5UVUXk