Well, it's kind of complicated, but it has to do with grenades, Belgium, the Battle of Waterloo, and Eskimos. (Credit for a correct answer went to anyone who got either the Waterloo part, or the grenade part, correct.)
The big fuzzy bearskin caps are the traditional headgear of grenadiers (soldiers who throw grenades for a living). These caps are tall and imposing, but more importantly they have no brim to get in the way when one is lobbing grenades at one's enemies.
However, the job of the royal guards is to guard the monarch, not throw grenades, so the royal guards did not originally wear the big fuzzy hats - only grenadiers were allowed to wear bearskin caps.
Interestingly, The royal guards began at a time when there was no monarchy in England. Oliver Cromwell was in power, and King Charles II was in exile in the city of Bruges (in what is today Belgium). It was there in 1656 that he formed the Royal Regiment of Guards, later renamed the First Regiment of Foot Guards.
In 1815, this regiment fought at the Battle of Waterloo, where they defeated the bearskin-wearing Grenadiers of the French Imperial Guard. In honor of this accomplishment, the regiment was renamed "The First or Grenadier Regiment of Foot Guards". A somewhat awkward name, but with it came the right to wear the bearskin caps. This tradition has been extended to all five regiments of Foot Guards which today share ceremonial guard duties. In addition to guarding the Queen, these regiments are "real" soldiers, taking part in a variety of more traditional military roles, most of which do not require big fuzzy hats.
The bearskin cap itself is 18 inches tall, weighs one and a half pounds, and is made of the fur of the Canadian black bear. It is often confused with a "busby", which is a significantly smaller fur cap. In military circles, calling a bearskin cap a "busby" is something of an insult. (Apparently size matters.)
The bears are not killed specifically to make headgear. The Canadian government allows periodic culling by Inuit (Eskimo) hunters, to keep the bear population under control. Still, the folks at PETA are not happy about this, preferring the bears starve to death naturally, rather than being killed by humans and subsequently worn on their heads. The Ministry of Defense, not totally immune to their pleas, has been working for a number of years to replace the bearskins with synthetic fur. So far no replacement has been found with the same performance and durability as the natural bearskin. (It seems the synthetic bearskin caps exhibit some interesting problems with static electricity.)
(Bob - I suppose that could be where they hide their bottle of Gordon's Gin.)
(Allan - And yes, I do believe that a bearskin cap would in fact keep a soldier's head warm.)
WHO GOT IT RIGHT: Bob Milligan, Robin Campbell, Marc Quinlivan, Jim Ahumada, JP Weigt, Allan Christensen, and Pearl.