The service used phone lines connected to Electrophone's special receiver. All you had to do was call them up, and a phone operator would ask what you wanted to listen to. If the user wanted to listen to a sermon, the call would be redirected to a church where microphones were installed (sometimes disguised as Bibles, because for some reason microphones were a point of contention in the Lord's house). For an opera, the operator would connect the call to the Royal Opera House, where another set of special microphones would stream the show.
"I'd like to hear the Queen's bath time ambiance."
"Right away, sir."
Of course, this cutting-edge technology didn't come cheap. The subscription cost five pounds a year -- the equivalent of around $570 today. That's five and a half Amazon Prime memberships. However, adding an extra receiver to an existing line only cost one pound, which allowed the wealthy-but-not-that-wealthy of the era to pool their resources into galleries with multiple Electrophone receivers. It was the 19th century's version of sharing a friend's Netflix password. The company even provided coin-operated machines, so people could pay per listen to keep up with their favorite sermon, rather than spring for the whole subscription.
Surprisingly, it was way more economical than modern cable television services.
Electrophone wasn't the only player in the game, either; other countries had companies providing similar services. Hungary had Telefon-Hirmondo (literally the Phone Newspaper), and France had Theatrophone way back in 1881, which came with Victor Hugo's seal of approval. We assume his endorsement was their equivalent of those DirecTV ads starring Rob Lowe.