Like most career-minded folks, I've never been willing to say "I'm happy where I'm at, and I'm not interested in further career growth". With that in mind, I've had to constantly examine my limits, my talents, and my passions to figure out my direction.
My limits: I love Computer Science. I got my BS and MS degrees in Computer Science. But I don't identify myself as a "Computer Scientist." That, to me, is a much more intense title. I don't have the patience or the mental capacity to solve the hardest problems in my field. I find complex problems like search algorithms, frameworks, or massive scalability to be intellectually fascinating, and I love learning how they were solved, but I'm not the guy who's going to solve them myself. This is why I chose not to follow a purely technical ladder beyond the level of Senior Software Engineer, or go for a PhD.
My talents: What I bring to the table is a healthy mix of technical and architectural skills coupled with organization, communication, and interpersonal skills. I have a strong sense of the "keep it simple" design philosophy, and I'm unlike many engineers in that I see value in and am good at writing clear specs, diagrams, e-mails, plans, and other forms of engineering beyond code. As I like to say "Even a simple e-mail has a user experience that needs to be considered." Also, over time, I've gotten good at paying attention to more work going on around me, but at a higher level. This is a key function of a manager, but not so much a requirement for an engineer.
My passions: I a nutshell, my passion is for seeing ideas turn into applications in an elegant way. As an engineer, I got to do just that pretty much on my own, and it was rewarding. But I was often frustrated that the coding process just took too long. As a manager, I get to do it with leverage. I get to be involved in more projects coming to fruition. As long as I'm involved, I don't feel the need to be the actual coder. Quite often I'm heavily involved in conceptualizing requirements and creating high level design (like, for instance, the data model), but then leaving the coding to one of my engineers.
Anyhow, to answer the question, my transition to management happened in my current job, where I hired on as an engineer six years ago and gradually became more and more of a manager. Yes, there was a specific moment early on when my job title changed, but the mental transition was much more gradual, and in some sense is still not finished. With a team size that has grown from 2 to 9, my hand was somewhat forced, and I've had to learn to resist the inclination to just "do it myself." Its been a good learning experience.
At one point a couple years in, my boss said to me point blank "I value your management skills much more than I do your technical skills." That was a real eye opener, since at that point I still identified myself as an engineer, and a good one at that. It was hard not to take it as a put down.
But I've come to realize that good engineering skills are a required part of a bigger package that makes a good engineering manager. At this point in my career I'm cultivating the rest of the package.