What is the history of the English Beefeater's tall hat? Why is it so tall and fuzzy, and how can that serve a purpose in battle?

Beefeaters do not wear tall fuzzy hats, the Buckingham Palace Guards or Household Troops do. They are called Bearskin Hats. The Yeomen of the Guard, or Beefeaters, are ceremonial guards at the Tower of London who wear scarlet and gold uniforms from the fifteenth century and carry a halberd. As for the history of the hats themselves, see:
https://en.wikipedia.org/wiki/Bearskin

The tall fuzzy hats are known as bearskins. They are not worn by Beefeaters but are in fact part of the ceremonial uniform of the Foot Guards, who act as the personal troops of the Sovereign with the special privilege of guarding the Royal Family in both peace and war. The five regiments of Foot Guards are the Grenadier Guards, Coldstream Guards, Scots Guards, Irish Guards and Welsh Guards. Their main ceremonial duties include the Guard at Buckingham Palace, St James's Palace, Windsor Castle and the entrance to the Tower of London. But the Foot Guards fulfill the unique dual role of being operational infantry troops as well as carrying out their ceremonial duties. They would not therefore wear the bearskin on combat duties.

The Yeomen of the Guard are entirely separate from the Foot Guards. They act as a royal bodyguard to the Sovereign on most State functions. They are in attendance at the Coronation, lying-in-state and funeral of the Sovereign, the Investiture of the Prince of Wales, the State Opening of Parliament (including the traditional searching of the vaults), the Royal Maundy Service, all investitures and garden parties at Buckingham Palace and also at the installation of Knights of the Garter at Windsor. Their headquarters is at St James's Palace where they keep their colour (flag) and their uniforms. They have a unique Tudor uniform consisting of a scarlet doublet embroidered with a Tudor crown and the roses of York and Lancaster. They carry a sword, which is not drawn, and a seven-foot halberd known as a partisan.

Although similarly attired, the Yeoman of the Guard are not the same as the Yeomen Warders at the Tower of London, whose duties are confined to guarding the Tower and its occupants. Yeoman Warders live with their famililes within the walls of the Tower and have no duties beyond the Tower precincts apart from forming a guard of honour inside the annexe at Westminster Abbey on the occasion of a coronation.

The Household Cavalry is the mounted Household Troops of the Sovereign and consists of the two senior regiments of the British Army ie The Life Guards and The Blues and Royals. The Household Cavalry provides mounted escorts for the Sovereign and the Royal Family on State occasions in London and also mounts the Queen's Life Guard daily at Horse Guards in Whitehall, London. On ceremonial duty the troops wear helmets and cuirasses and are mounted on black horses. The Life Guards have scarlet tunics and white plumes on their helmets; The Blues and Royals wear blue tunics and red plumes. Both regiments are fully operational armoured units liable to be stationed anywhere in the world. As far as I am aware bearskins are not and never have been worn by the Household Cavalry.

Though the use of bearskins in London today is purely ceremonial and traditional, the wearing of tall hats in battle has been used by many peoples around the world as a form of psychological warfare. The warriers appear taller than they really are in order to intimidate. This, together with war cries, rhythmic beating on shields, etcetera is simply done to weaken the morale of the other side.