What is the purpose of the white half-moons on the bases of our fingernails and toenails? Why don't they grow out with the nails?

Those white moons are called lunulae. The lunula is the only visible portion of the nail matrix, which produces the nail it self. The matrix (and the lunulae) never moves, but new nails continually push forward, away from the matrix.

Why does a lunula appear white? Dermatologist Harry Arnold explains:

'The nail beds distal to the lunulae look pink because capillaries with blood in them immediately underlie the nail plate. The lunulae look white because the thin, modified epidermis of the nail bed is three of four times thicker there, being the busy factory where nail plate is manufactured. The lunula is avascular [without blood vessels], so it looks white.'