Taxpayers paid £2.4 million to renovate the Grade-2 listed building, which the couple moved into just nine months ago.
Work on the property in Berkshire overran and it was reported that the couple made constant design changes, meaning the builders fell behind schedule.
Their new website sussexroyal.com says Harry and Meghan moved to Windsor for "various reasons", and mentioned the fact that their previous residence of Nottingham Cottage on the grounds of Kensington Palace "could not accommodate their growing family".
"As a result, Her Majesty The Queen offered the Duke and Duchess the use of Frogmore Cottage, which was already undergoing mandated renovations, and would be available to move in before the birth of their son.
"The refurbishment cost equated to 50% of the originally suggested property for their proposed official residence at Kensington Palace.
"It is for these reasons, the Duke and Duchess of Sussex chose Frogmore Cottage as their official residence," the website says.
Crucially, the website points out that Frogmore Cottage is owned by the Queen, therefore it will be up to her whether they will be permitted to use it as their UK home going forward.
The website says: "Frogmore Cottage will continue to be the property of Her Majesty the Queen.
"The Duke and Duchess of Sussex will continue to use Frogmore Cottage - with the permission of Her Majesty The Queen - as their official residence as they continue to support the monarchy, and so that their family will always have a place to call home in the United Kingdom."
Despite having "cottage" in its name, Harry and Meghan's UK home - located close to Windsor Castle - has four bedrooms and a nursery.
Frogmore Cottage underwent major work to turn five properties back into a single mansion - with all fixtures and fittings paid for privately by the duke and duchess.
The couple are said to have hired former Soho House interior designer Vicky Charles to transform the residence.
It is likely that they installed a luxury kitchen and bathroom, and it has reportedly been designed by Harry and Meghan with dining and entertaining in mind and with extra bedrooms to accommodate guests, such as the duchess's mother Doria Ragland.
After annual royal accounts were released last June, campaigners called for a parliamentary inquiry to be launched into spending on the monarchy.
Republic, which calls for an elected head of state, questioned why £2.4 million of taxpayers' money had been "thrown" at Frogmore Cottage while public services were under financial pressure.
Frogmore Cottage, which is about half a mile south of the Queen's favourite residence Windsor Castle, is close to Frogmore House where the duke and duchess held their evening wedding reception.
Frogmore House is also where the couple posed for their official engagement photos.
It has been a royal residence since 1792 and was built in around 1680 by Charles II's architect.
In 1923, the Queen's parents - the future George VI and Queen Elizabeth - spent part of their honeymoon there.
The name Frogmore is derived from the frogs which have always lived in the low-lying marshy area.