[alancalverd (OP)]

Or, for that matter, any fruit?

The skin of an apple is tough, inelastic, and quite different from the pulp. Apples obviously grow from the inside (the stalk is the only source of material) but you rarely see one split or burst, so some material must migrate from the interior to the surface and change from white to green on contact with the air, then ripen to red. But if you cut an apple, the pulp oxidises to brown without forming a new skin, so the skin cells are functionally different from the pulp. And the clever thing is that the migration is almost isotropic - the apple just gets bigger without altering its aspect ratio.

It's an essential, everyday phenomenon, but what's the mechanism, please?

[Bored chemist]

The cells on the outside of the apple divide creating more "apple peel" cells.

[evan_au]

It is due to epigenetics, where different cells tag certain genes as active, and others as inactive,
- Skin cells will have a different set of genes enabled than the pulp or or the stem (even though they have the same genes)
- The fertilised seeds have different genetics than the apple itself, since half of the seed genes are derived from the pollen (often from another tree)

It's the same mechanism that occurs in you and me during development.

[Petrochemicals]

Then same as the bark of the tree. How does a tree add girth to itself when he outside is dead ?