The name the Royal Mews, despite being first and foremost a stable for the royal family, derives from its use as a home for the family's hunting falcons and hawks. "Mews" is a term for the time of year when the birds of prey shed their feathers, and now, as you were thinking, the noise a pretty kitten makes.
The Royal Mews moved to this spot in 1760, when King George III decided to relocate to Buckingham House. When George IV took the throne in 1820 and set about converting Buckingham House into a palace, he commissioned John Nash to rebuild the stables with grandeur. Further improvements were made to the Mews when Queen Victoria moved into Buckingham; Prince Albert had a new forge inistalled and additional cow sheds built onto the Mews. You can never have enough cow sheds in Central London.
In 1855, Queen Victoria used her own money to establish the Buckingham Palace Royal Mews School, which served to educate the children of the stable's servants. The Mews had nearly 200 employees at the time, and accommodations were built for them and their families in 1859.