Lemon juice is an acid, so it makes the cut surface of the apple more acid. Enzymes often turn fruits brown, and enzymes usually work best when there's not too much acid around them - or too much alkali [base]. Then the pH is Neutral.

Baking soda is a mild alkali [base]. Does baking soda keep the cut surface of an apple from turning brown?

The acid in the lemon juice prevents the oxidization reaction that causes the apple to turn brown. I don't know further details, but you can do this with other acids.

The reason an apple turns brown when it is exposed to the air is because cutting open the cells of an apple causes them to release an enzyme called polyphenol oxidase. When it reacts with the oxygen in the air, it causes the apple to turn brown. An analogy would be how iron will rust when exposed to oxygen (although the mechanism is slightly different).

Adding lemon juice does two things:
1. the Ascorbic acid in lemon juice will preferentially react with oxygen compared to polyphenol oxidase. To continue the metaphor from above, ascorbic acid is like the zinc coating on stainless steel which prevents the iron in the steel from rusting.
2. lemon juice has a very low pH of about 2 (that means it is very acidic). Most enzymes, including Polyphenol oxidase, only work well within a certain range of pH. For polyphenol oxidase, this range is about 5-7. Therefore, by lowering the pH at the surface where polyphenol oxidase is being released, the lemon juice helps slow browning.