Why do my egg whites separate after whipping?
This is the nature of meringue: they will start to fall apart as soon as you stop whipping.  There are a few tricks to help it hold longer, but in general you want to have EVERYTHING ready to go as soon as the meringue is whipped.

To help stabilize the meringue you can:


Use a copper or SILVER-plated bowl to whip, or add a tiny amount of powdered copper supplement from a health food store
Acidify it slightly: add 1/8 tsp cream of tartar or 1/2 tsp lemon juice per white, before beating
Let the bowl warm to room temperature, which increases the ability of the whites to take in air
Ensure there is absolutely no yolk in with the whites.  The fat greatly destabilizes the foam.


Now for WHY these tricks work:

I'm going to quote heavily from Harold McGee's "On Food and Cooking", as it does a wonderful job explaining meringues and other whipped egg whites: 


  Like the head on a beer or a cappucino, an egg foam is a liquid--the
  white--filled with a gas --air-- in such a way that the mixuture of
  liquid and gas keeps its shape, like a solid.  It's a mass of
  bubbles, with air inside each bubble, and the white spread out into a
  thin film to form the bubble walls.  And the makeup of these liquid
  walls determines how long a foam can stand up.  Pure water has such a
  strong surface tension--such strong attractive forces among its
  molecules--that it immediately starts to pull  itself together into a
  compact puddle; and it's so runny that it puddles almost immediately. 
  The many nonwater molecules in egg white both reduce the surface
  tension of the water they float in, and make it less runny, and thus
  allow the bubbles to survive long enough to accumulate in a sizeable
  mass.  What gives the mass of foam a useful kitchen lifetime is the
  white's team of proteins.


Whisking unfolds these proteins, primarily globulins and ovotranferrin, which bond to each other and stabilize the bubble walls.  Cooking will evaporate the water and unfold ovoalbumin, creating a rigid and permanent protein network.  

However, the same proteins can ALSO destabilize the foam if they bond too tightly. "The protein network begins to collapse when too many of these bonds accumulate and the proteins cluster together too tightly" (page 102). In the case of egg proteins, one of the strongest bonds is a disulfide bond between the sulfur-containing amino acids, cysteine and methionine. Eggs contain copious quantities of these amino acids, which are why they produce such a potent stench when they spoil; the sulfur is converted to malodorous sulfur compounds, particularly hydrogen sulfide.  

Copper, silver, and acids stabilize the egg foam by preventing the formation of these disulfide bonds. To quote Harold McGee (page 103): 


  It turns out that along with a few other metals, copper has the useful
  tendency to form extremely tight bonds with reactive sulfur groups: so
  tight that the sulfur is essentially preventing from reacting with
  anything else.  So the presence of copper in foaming eggs whites
  essentially eliminates the strongest kind of protein bond that can
  form, and makes it harder for the proteins to embrace each other too
  tightly.


McGee also notes that silver has the same property of inhibiting disulfide bonding. Acid achieves the same goal of reducing disulfide bonding, but works slightly differently:


  The sulfur bonds form when the sulfur-hydrogen (S-H) groups on two
  different protein molecules shed their hydrogens and form a
  sulfur-sulfur (S-S) connection with each other.  The addition of an
  acid boosts the number of free-floating hydrogen (H) ions in the egg
  white, which makes it much harder for the S-H groups to shed their own
  H, and so slows the sulfur bonding down to a crawl.