When Sherlock was a child, he idolised his older brother Sherrinford, he was always kind to him, and had a cool job (something like an MI5 agent, where there's lots of danger and secrecy). They'd play spies together, and sometimes Sherr would like Sherlock wear his coat, even though it was far too big for his brother.
Sherlock was crushed when his brother died young, a mission gone wrong or something, he didn't get to say goodbye or anything. So when he saw the Belstaff in a shop window he had to have it, so he could have that reminder of his brother everywhere he goes.