"I was emotional, scared and nervous. It was absolutely unbelievable," said Smith, a Commonwealth Games gold medallist for England in 2006 and world bronze medallist last year.
"I thought it would feel like competing in a Commonwealth Games or World Championship, but it was a completely different level.
"I hope this medal is a stepping stone for British gymnastics. I hope it will inspire youngsters to come into our sport."
Video - Smith speechless after historic bronze
Walter Tysal won Britain's only previous individual men's gymnastics medal when he took silver in the all-around competition in 1908.
In 1928, in Amsterdam, the British women won team bronze.
Xiao, the three-time world champion, led throughout to claim China's fifth gymnastics gold of the Games.
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