I do not understand what happens in Villette after Miss Marchmont's death, when Lucy Snowe is subject to an Aurora Borealis.

In the novel Villette, Bronte often uses the parallel of unusual or disturbed weather to symbolize a coming momentous change for her characters.  By the same token she employs the use of calm or happy weather to reflect the emotions of a happy scene.  In Chapter V "Turning a New Leaf" Lucy Snowe, having recently left her brief employment with the newly dead Miss Marchmont, is walking to "consult an old servant of our family; once my nurse, now housekeeper at a grand mansion not far fom Miss Marchont's" (47).  Lucy Snowe is alone in the world, and has lost her sole possible resource (Miss Marchmont or a legacy from Miss Marchmont), and is quite frightened to be thus completely thrust upon the world.  Her heart is in such an extreme state of upheaval that the appearance of the Northern Lights (aurora borealis) lead Lucy to think these somewhat fanciful, if understandable, thoughts:

Not feebly, I am sure, or I should have trembled in that lonely walk, which lay through still fields, and passed neither village nor farm-house, nor cottage; I should have quailed in the absence of moonlight, for it was by the leading of stars only I traced the dim path; I should have quailed still more in the unwonted presence of that which to-night shone in the north, a moving mystery -- the Aurora Boreaslis.  But this solemn stranger influenced me otherwise than through my fears.  Some new power it seemed to bring.  I drew in energy wih the keen, low breeze that blew on its path.  A bold thought was sent to my mind; my mind was made strong to receive it. 
     "Leave this wilderness" it was said to me, "and go out hence."
     "Where?" was the query.  (Ibid)

Specifically, what Lucy feels is happening is that the very energy of the Northern Lights is transferred into her, giving her strength and resolve to strike out on her own.  Northern Lights appear as moving sheets of colored light in the northern sky, and are caused by solar energy.  It is not unusual that a young woman, completely alone and friendless, might take this as a sign of courage and resolve sent to her by some higher power.  Lucy doesn't refer to God specifically in this passage, but she seems to believe in the power of her natural surroundings to influence her mood and actions.  Lucy had just lost Miss Marchmont on a horribly stormy night (in Chapter IV), so it seems likely that Lucy sees a connection between weather events and human feelings and decisions.   It is at this moment that Lucy makes the decision which will lead her to travel away from England to the town of Villette in Labassecour (Bronte's code word for Belgium.)

Source: Bronte, Charlotte.  Villette.  New York: The Modern Library, 2001.