You've probably heard that oil and water don't mix, but milk is in fact a mixture of many different components, including sugars, proteins, fats and yes, water.  [Fats and oils are basically similar molecules, also often called lipids, and made of smaller building blocks called fatty acids.  Although fat, oil and lipid are often used to refer to fats, "oil" generally refers to a fat that is liquid at normal room temperature and "fat" is solid at room temperature.]  The oil or fat in milk can mix with water without separating because they form into little globules sort of like water balloons.  The fat is inside a thin skin or membrane which helps attract water somewhat, and because fat is less dense than water, the globules float and form what's called a colloidal suspension or emulsion.  The smaller the globules, the easier it is for them to remain emulsified, but if they are too large they will separate and float to the surface.  Years ago before milk was homogenized, the fat globules would float to the top and form a layer called the cream, which contained most of the milk's fat.  Homogenization breaks up the larger globules into smaller ones, which, along with the milk proteins that then help form a stronger globule membrane, prevent this separation.  The cream you buy at the grocery store is separated at the dairy and then homogenized to keep the fat suspended in the liquid.  Heavy whipping cream means that it has a very high fat content, 36-40% (normal cream is about 30% fat, while Half & Half is 12% and whole milk 3.5% fat).  Butter is essentially pure milk fat.

In our heavy cream, the fat globules are still emulsified, or suspended in the water along with some protein and sugar.  If we shake the cream though, the globules will begin to crash into one another and pop or burst their membranes.  When this happens the fat molecules spill out and clump together with other fat molecules.  At first as this happens the fat traps a lot of air and forms bubbles or foam, which is the whipped cream.  The more you shake it, however, the more bubbles pop and the more densely the fat clumps become until they can no longer remain suspended in the water, and they separate completely.  This is the butter, and the remaining liquid, mostly water and proteins, is called buttermilk. 
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The butter you make is not only good to eat, but will stay fresh for a long time, especially if it is thoroughly rinsed of all the milk liquid, since the milk can spoil.  Adding sugar to the whipped cream not only sweetens and thickens it, but helps prevent the fat bubbles from popping so that it will stay a light fluffy foam much longer.