"Closet" in this sense simply meant a private room (from the Old French "clos", where we also get "closed" in English), of which a palace would have several. Whitehall and Greenwich palaces both had several "closets" for both the King and Queen, including writing closets, robing closets, and general closets.

To understand the role of the closet as a room we need some context on Tudor royal palace rooms: different rooms would be more or less private, with the most private (the bedchamber) reserved for closest courtiers, friends, advisors, and; in the case of Catholics; confession takers. After this was the robing closet, where one might be dressed informally, but was still only for very close courtiers or closer. Then came the closet, for general "private" functions (note: in this context "private" means more "not state" than secret, and here we drift into the idea of monarchs as dualities - regent and person, which is another question!)

So, in short, this simply means it was a small ceremony behind closed doors.

There were also specialist closets, such as writing closets, for specific functions - which would be restricted to different groups.

Architectural curator here. Closet was a term used for a small-ish room in a set of apartments in any large dwelling from the late medieval era through the early 19th century. This was particularly true of European estates. The closet was frequently a private room off a bedroom suite. Abigail Adams, wife of future president John Adams, was particularly taken with a closet in one of the houses they rented while John was serving in France. With a window and little desk, she thought it was a nice quiet space in which to write letters. The modern closet evolved from that concept in working class and middle class houses, but the Industrial Revolution and the resulting changes in social and domestic arrangements destined the modern closet space for storage of the many consumer goods now available to the masses in the 19th century.

In this period, one often reads about someone "hearing Mass in her closet", which should give you a better idea of what this means — a small area of the apartment, large enough for a religious ceremony, but private enough that only a handful of people could be there.

Among royals of the Tudor period, one of the greatest luxuries that they could experience was privacy — in virtually every moment of their lives, they were surrounded by people. While getting dressed, attending religious services, and notoriously in Henry VIII's case, moving their bowels, they were aided and assisted by courtiers, who would both attend to the needs of the monarchs in question as well as offer their opinions on the issues of the day … and ask for favors for cousins, or a signature on a petition for a grant of land, or a remission of tax, etc.

It is important to realize that the only persons properly reared to royal duties in the Tudor family were Elizabeth of York, Henry VIII's mother, and Catalina of Aragon, Henry VIII's first wife. Henry VII had grown up in obscurity and exile, making multiple attempts at seizing the crown with mercenaries before succeeding against Richard III. His wife Elizabeth of York, on the other hand, was the eldest daughter of Edward IV and had been reared at court, and was at the court of her uncle Richard III as well. She appointed her cousin, Margaret, daughter of George of Clarence, her paternal first cousin, to rear her firstborn son Arthur when he took his patrimony of Wales. Henry VIII was a second son and Margaret Beaufort, who invented the title "The King's Mother" for herself, was famous for her piety and learning, and he was reared with a view to serving in the Church, and only came to his role as heir very late, as his brother survived until his teens.

Anne Boleyn was from a family of county gentry, though her father had managed to have her reared at courts in the Netherlands and France. Jane Seymour was from a large county family, and may have considered taking the veil as a nun, and at any rate, was notoriously unworldly. Anna von Kleve was from a German duchy, and any polish and poise she might have had were nearly impossible to see given the language barrier. Katherine Howard was from an important family, the Howards that held the Duchy of Norfolk then and do now, but was an extremely minor child and had been neglected greatly during her childhood. Catherine Parr was wed twice before her marriage to Henry and was the experienced chatelaine of a northern lord, but was still comfortable in another element.

While we might see an elopement or a very private wedding ceremony as a sign of impulsive decision-making or even shame, realistically very few people were prepared for the enormous burden of existing with dozens and hundreds of personal attendants, all of whom wanted something from them, and all of whom had a personal agenda to be considered. A private wedding could be a gesture of consideration to a bride in this case.