So if The Police were a punk band originally, where did the idea to play reggae come from?

Drummer Stewart Copeland explains in his memoir Strange Things Happen:

“At the punk clubs, they would play punk music.  But you can’t play punk all night without giving everybody headaches, even punks.  So they had to have some variety in the energy levels.  But there’s no such thing as mellow punk.  So they would alternate the punk music with reggae, particularly dub reggae.  

“This got all the punk bands thinking, ‘Damn, we should try to play that stuff.’  But none of them were very good at it.  The Clash, The Damned, The Sex Pistols—all those groups had great charisma and great enthusiasm, but The Police just were better at it than anyone else!  We were better at just about everything than anyone else because we were real musicians.”