Key source of soft power
The British Monarchy is a big influence in how the world sees Britain and how we interact with other nations.
Tourism is a large industry in the UK, and Royal destinations are some of the most significant tourist sights, e.g. Buckingham Palace, Windsor Castle
Many people are attracted to the UK for its history. The Royal Family is an important part of that, and many people want to visit the places they live. There is a special appeal in that the Royal Family is not just historical, but still functioning today. The British monarchy in particular is famous as being one of the most longstanding, and is famous the world over. This means many people come to the UK and spend money because of the Royal Family, which employs people and contributes to the economy significantly. Abolishing the Royal Family would mean a large drop in tourism, as one of the countries most unique and interesting attractions would be gone.
Many of the things people come to see could exist without the Monarchy, such as Buckingham Palace and the changing of the guard. If the Royals were no longer in residence, these could be opened more and generate more revenue.
The Royal Family generates money and business for the UK economy. We should keep things that benefit the economy.
Economic benefit alone is not a reason to keep something if it causes other problems.
Content references here ...