The big problem for me had always been speaking anxiety. I was terrified to get up in front of people and talk. And then, suddenly, I was not. I have no idea how to induce this change in others, but I assure you that it made a far bigger improvement in my speaking style than any change in intonation.
If you want to do what the article says, just listen to someone British for a while and then try reading some words aloud, enunciating carefully. Try to be as "accentless" as possible, whatever what may mean in your area. This becomes habit after a little while.
It's entirely about experience, and the threshold for each person is different. And as experience goes, there's always new challenges. You might not be nervous now, but I bet if you got put in front of a large enough crowd (think Obama-sized) it would come back. But even then it's about experience, you learn to make it work for you.
If you want to do what the article says, just listen to someone British for a while and then try reading some words aloud, enunciating carefully. Try to be as "accentless" as possible, whatever what may mean in your area. This becomes habit after a little while.