It's a little bit hard to describe, but I suppose "It turns out" is kind of another way of saying "I found out" or "it was revealed that", or something similar. So another way of saying the same sentence could be: "I found out the next day he spoke to the coworker and told him everything we had discussed." We normally use "it turns out" when expressing surprise at what we found out. For example, say there's a movie where someone was killed, and you find out that the murderer was the person's wife, and it surprises you. When telling your friends about it you might say: "It turns out it was his wife all along!" It's probably one of those phrases where the more you see it used, the more you'll start to understand how it's used/what it means. You'll probably hear it a lot as we use it quite often.

to turn out - to prove to be. E.g., Her work turned out to be badly done (when we checked it). It means you have an opinion about something but then something happenes and you realize that it is not like that. "Turns out the next day he spoke to the coworker and told him everything we had discussed." - You think he will not tell it to anybody but then it turns out (this fact comes to the surface) that he spoke to the co-worker.