Why Does Lemon Juice Keep Cut Apples Fresh?

It's the acidic level in the lemon juice that does it. Works on banana's too.
Not sure why though, but it is really good to drop sliced banana's into a bowl of lemon juice before putting them into a banana pudding. Keeps them from turning all brown and yukky looking.

The acid denatures ("unties") the protein's complex structure.  There is a protein in the apples called polyphenol oxidase that reacts with a particle in the cells of an apple.  When you break the cells (by cutting the apple or dropping it) these two things react together and create a substance called melanin, which is a brown or black pigment.  This is why the apple turns brown.  The acidity of the lemon juice breaks apart the polyphenol oxidase and prevents it from being able to react with the stuff found inside the cell  (called polyphenolic substrates) and as a result melanin can't be produce which equals no brown colour!