Older cats lick new kittens mostly for practical reasons, but the pleasant sensation of being groomed is a perk that most kittens enjoy. For both practicality and as a sign of love, then, a kitten may grow up into a cat that enjoys licking even her human owners. When an older cat licks a kitten, it's because of evolutionary necessity, and it's a behavior long-ingrained in their instincts.
Licking at Birth
Kittens are relatively helpless at birth, and if their mother doesn't lick them, they may not survive the birthing process. When a kitten is born, he's covered in afterbirth fluids, and his face may even still be covered by the amniotic sac. The mother licks her baby off, clearing away any obstructions near the face and stimulating him to breath in the open air. She may need to lick and nudge them toward her to encourage them to nurse.
Older cats lick new kittens after a meal for two reasons, the first of which is that these are meticulously clean animals, and kittens can be messy eaters. The bigger concern, however, is going to the bathroom. A kitten can't poop on his own until he's two or three weeks old, so the older cat will lick his hindquarters to stimulate it. Because of this, people raising orphaned kittens must mimic the activity and stimulate their kittens to go to the bathroom by rubbing a kitten's anal area with a warm, moist washcloth.
Marking for Scent
Scent is an important way for cats to understand their surroundings, and especially to claim their territory. Cats mark their territory using scent, either by scratching to leave behind scent from their paw pads, by urine spraying after sexual maturation or by licking. Licking is how an older cat marks her young and identifies them as hers. When they start to lose her scent -- if they become lost and wander from the nest, for example -- she'll lick them as a way of re-marking them. This is also a reason that cats sometimes lick their owners, as it establishes a sense of mutual ownership.
Cats don't lick new kittens just for practical reasons -- it's also a valuable bonding exercise that keeps the relationship close. Kittens need to be groomed, of course, but the act of meticulous and rhythmic licking with a mother's warm, rough tongue helps establishes a close and loving bond. This is the other reason that a cat may also lick his humans -- he learns at a very young age that it is an exercise deeply rooted in affection and care, and that the feeling of being licked is pleasurable.