As per the comments - 'run about' is the same as 'run around'. It's not really a phrase, it's just two words - 'run', and 'about'. 'About' means something like 'in different directions'. The dictionary meaning you quoted is just an extension to that basic idea - If you are busy, you have to figuratively run from place to place to get things done.

There is an interesting thing, the British usually say that, but I find that American say run around.

I am not familiar with this phrase, we usually say "run around," instead.

thanks~

i guess that means: to move busily from place to place. http://dictionary.reverso.net or a person who roves around from place to place. http://www.thefreedictionary.com