Can someone explain how royal titles work. why's Prince William now Duke of Cambridge after his marriage?

Their full titles are "Their Royal Highnesses, Prince and Princess William, Duke and Duchess of Cambridge, Earl and Countess of Strathearn, Baron and Baroness of Carrickfergus". The ducal title is of higher status than being merely "Prince and Princess William," since it is a royal peerage, so it is the one used officially. William is still a prince, and Catherine is a princess-by-marriage (all royal duchesses are), but she is not a princess in her own right.

Only women born princesses can use the title before their own given names (though it's true that an informal exception was made for the late Duchess of Gloucester, who was permitted when she was elderly and widowed to call herself Princess Alice). All others must use a style indicating that they derive their titles from their marriages. Hence "Princess William" and "Princess Michael of Kent".

Prince Edward's wife, the Countess of Wessex, also holds the title "Princess Edward".

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