Why doesn't the cell membrane just...break apart?
The membrane bilayer is held together by hydrophobic forces. This is an entropy driven process. When a greasy or hydrophobic molecule is suspended in water, the water molecules form an organized "cage" around the hydrophobic molecule. When two hydrophobic molecules come into contact, they force the water between them out. This increases the entropy because the freed waters don't need to be organized into the cage. Lipid bilayers have many many many hydrophobic lipids that squeeze out a lot of water and greatly increase entropy. The polar phosphates allow the water to interact with the surface of the membrane, without a polar head group the lipids would form a spherical blob instead of a membrane.

Read this section on wikipedia for more.