What was the original purpose of such a shirt, I wonder?
Extra warmth against the wind, I believe. Probably caught on more because it was a military style, though (it's like the reversible plastron-front of a lancer's jacket), and people thought it made them look cool. I've often seen them referred to as "fireman shirts" too, implying they were a popular uniform for early fire departments. Again, probably because they were military-looking.