Broadly speaking, The King's Speech is a movie about a guy who needs to do a very important thing but can't, because he has a stutter. Then the first guy meets another guy who helps cure his stutter, allowing him (the first guy) to do the important thing he had to do, because he gained confidence.

This does not seem like the outline for a film meant to prop up the thousand-year old British monarchy. But what if we told you the first guy was King George VI and the important thing he had to do was read England's 1939 declaration of war against Germany on live radio? Suddenly, the dots begin to connect. Marketed as something fans of movies about lady boxers and cannibalistic Uruguyan rugby players overcoming adversity could enjoy, The King's Speech is one long commercial for the virtues of the British monarchy, writes The Guardian's Steven Fielding. Seen from this perspective, the picture is nothing but "another addition to the royal family's filmography, a vast body of work that has played an underappreciated but insidious role in maintaining support for the monarchy."

Fielding continues:

The royal family – always the worst judges of their own self-interest – initially discouraged the production of dramatisations of themselves by making their displeasure known to those censors who vigorously policed stage and screen. In 1937 the lord chamberlain even issued a formal ban on the portrayal of sovereigns on the stage until a century after their accession. This was undoubtedly meant to prevent dramatists writing about the recently abdicated Edward VIII. It did, however, mean that depictions of Queen Victoria could be shown on stage for the first time, as she came to the throne in 1837. ... British cinema had been craven in its support for the status quo, but was keen to take advantage of this more liberal attitude....These movies--which could not have been made without royal co-operation--helped out by emphasising Victoria's selfless dedication to promoting her people's interests
...
The King's Speech is part of this established pattern. It could not have been nice for George VI to have a stutter and George V was not the world's most loving dad....[but] the film does its best to draw a veil over royal support for appeasement. In 1938 the King and Queen appeared on the balcony of Buckingham Palace with Neville Chamberlain after his return from Munich having sold out the Czechs. Now why is that, rather than a speech impediment, not worthy of a film?