In the early 1800s, Arthur Wellesley, then Viscount Wellington, asked his shoemaker to make boots in this style because they were easier to wear with trousers. After his victory over Napoleon at the Battle of Waterloo in 1815, the boots achieved the status of must-have fashionable footwear and duly became known as 'Wellingtons'. This particular pair is made of soft calfskin, polished black, with tan soles and spurs are fixed to the heels.
The Duke of Wellington spent 23 years at Walmer Castle and the bedroom where he died is preserved (more likely conserved and recreated) there for visitors to see when they visit.
Images of objects from Walmer, and more info, can be found on the English Heritage website.