I started by taking responsibilities for small teams and discrete portions of projects. From there, responsibilities expanded. Now I run a team of around 100 and focus most of my day on admin and training instead of actually creating work product. I like the role, but it's not for everyone. Managing is great when things are good, but can quickly get tough when things aren't.
Don't tell your boss you want their job. Tell them you'd like to have an opportunity to contribute in a greater capacity and how you can do that. If there's something specific that thrills you, ask to do that. Put 100% into making sure that thing gets done well.