Sunburnt skin starts peeling after a while because new, healthy skin is being formed to replace the damaged skin. This is why the upper or old layer of skin is shed after a person has been sunburnt. There is not much you can do about this, besides being disciplined enough not to scratch or pick at your sunburn in order to prevent scarring.
Do not exfoliate sunburnt skin, even though this is often advised. The new skin is still very vulnerable and prone to infection at this stage. Exfoliation will often cause small, minuscule wounds to develop and removes the new skin cells that are being produced to form fresh, healthy skin.
Be very careful when drying off after a shower and take short, lukewarm, almost cold, showers while the skin is still peeling. This is important advice to follow as water dries out the skin.
And we cannot say this enough: moisturize, moisturize, and moisturize! Your skin will thank you for it. Use a hydrating cream or oil that does not contain any aggressive ingredients such as perfume. Drink plenty of water as this is good for the healing process and the new skin.