The reason that the British Royal Family presides over the Commonwealth Games is because the monarchy are the rulers of the Commonwealth of Nations. The Commonwealth of Nations are the four main countries of Great Britian which are England, Scotland, Wales, and Northern Ireland. All of the countries, territories, and states that show adherence to the royal crown are also included in the Commonwealth of Nations.
The Royal Family takes great pride in being able to preside over the Commonwealth Games as everyone from Queen Elizabeth II to Prince Charles and Harry are huge fans of athletics with all of them being former athletes themselves. There is always at least one official member of the Royal Family who attends and presides over the Commonwealth Games. All of the athletes at the Commonwealth Games always make sure that they pay their respects to the Royal Family, especially if the ruling monarch is in attendance.
The CommonWealth of Nations contains former colonies of the "old" British Empire. Originally, when the CommonWealth was formed, it required the colonies who had received freedom to accept the King/Queen of England as the head of their respective nations (Canada still considers it!!). In 1949, when India desired to become a Republic without leaving the CommonWealth, the then King George VI asked the Parliament to come up with a solution.
That same year, the London Declarationwas passed. In this, the highest member of the British Royal Family would be considered Head of the CommonWealth and countries could choose whether or not to accept them as the Head of their country. As the Head of the CommonWealth, it is their duty to preside over the CommonWealth games(fun duties these guys have). If they can’t attend, other members of the Royal Family are required to act as the Head’s representatives. CommonWealth games in 2018 are being presided over by Prince Harry and Meghan Markle.