This is a song about sex with minors, and a casual attitude towards sex in general. "It's no hanging matter, it's no capital crime" means that there's nothing really wrong with it - that it's not worth hanging someone over.
The Rolling Stones used the sound of The Velvet Underground's "Heroin" for this song.
In the lyrics, the girl in the Beggar's Banquet version is 15 ("I can see that you're fifteen years old"). However, on their 1969 USA tour and subsequent ones, Mick Jagger reduced the age of the girl to 13. He wanted to test public opinion to the limit.
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