Jonathan H. Gutow of the chemistry department at the University of Wisconsin at Oshkosh provides the following reply to this oft-asked question:
"The short answer is that chemical compounds in the fruit are oxidized when the skin of the fruit--and hence the walls and membranes of the cells within the fruit--is ruptured, allowing oxygen in. These compounds react with oxygen, usually incorporating it into their molecular structure. Many of the resulting oxidized organic compounds have a brown color. Citric acid is very easily oxidized and can be used to scavenge oxygen to keep the fruit from turning brown. This is why, for instance, sliced apples will remain white for a much longer time if they are first dipped in lemon juice.
"A good reference for some of this is a book titled The Chemistry of Cooking, by A. Coenders (Parthenon Publishing Group, 1992; currently out of print but available in many libraries)."
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Judy Harrold is chief home economist for Ball & Kerr Home Canning Products. She fills in a few technical details
"Discoloration of fruit results when an enzyme called polyphenoloxidase oxidizes the phenolic compounds that are found in the tissue of fruits. The oxidation causes the phenolic compounds to condense into brown spots. While the tissue, or cell structure, remains intact, the enzyme and phenolic compounds are separated and so do not react with one another. It is the breaking down of the cell structure caused by the bruising that initiates the oxidation reaction and so triggers the onset of browning.
"For more information on this phenomenon, readers might want to look at the book On Food and Cooking: The Science and Lore of the Kitchen, by Harold McGee (Scribner, 1984). This is a very good reference."