Working class brass bands came about in the 1840s in Britain for a variety of reasons, but I will focus on the cultural and economic factors that drove the brass band movement in Britain.

Brass ensembles of various types have existed for hundreds of years, primarily in Italy and Germany. The modern British brass band emerged in Victorian England with a number of professional and private bands forming in the 1820s. Working class, amateur bands began to emerge in the 1840s. Miners and other British blue-collar workers were encouraged to participate in musical endeavors by a social elite who saw music-making as a respectable and worthwhile pursuit. Several early bands were founded and subsidized by industrialists; the Cyfarthfa Band in South Wales was probably the most famous example.

Why brass bands, as opposed to orchestras or wind bands? The newly-invented piston valve made brass instruments far easier to play than their keyed and natural (key or valveless) predecessors. This, combined with the emergence of mass production made brass instruments relatively affordable for the burgeoning British middle class.

It's worthwhile to note that brass bands enjoyed a similar popularity in the United States in the mid-19th Century. While a system of contests and competitions codified the brass band instrumentation in Britain, the brass band was eventually supplanted in the United States by the mixed wind band, popularized by Patrick S. Gilmore.

Source: my primary source is Trevor Herbert's entry in the New Grove Encyclopedia of Music, accessed through Oxford Music Online. My professional specialization lies in brass performance and pedagogy, but I have a pretty rigorous background in musicology. For more information, try these sources:

A.R. Taylor: Labour and Love: an Oral History of the Brass Band Movement (London, 1983)

A. Hailstone: The British Bandsman Centenary Book: a Social History of Brass Bands (Baldock,1987)

D. Russell: Popular Music in England, 1840–1914: a Social History (Manchester, 1987, 2/1997)