I’ve read today that one proofreader learnt copy editing while working for a publisher, comparing the copy editor’s notes with the author’s, giving them insight into the sort of things to look out for and how to do the job. I found this fascinating.
There probably isn’t a single way you can go about it. Some readers are particularly apt at spotting errors – or new angles of thought – in the books they read. Some, like myself, got started by helping author friends prepare for publishing. There are those who want the freedom that comes with running your own business, or for other reasons find working from home suitable.
Professional courses, colleague advice, and membership of professional organisations can help alongside experience. My view is that if you’re a writer getting your own writing edited or proofread it can help you see the value of the processes in action rather than just taking it for granted.