The literal meaning of that phrase is to have anxiety while being busy, in usual things, like life.
They always say that having some anxiety is normal and bla bla bla. But, if you have anxiety, then you have no idea what is normal, and you're waaay too busy worrying about whether or not it is normal, than to realize that it just is normal.
So what is an anxiously engaged person to do? Heck, I don't know, but I do know that this is what I do:
1) When Everything Else in Your Life is Out of Control, Control Things That Don't Matter so You Feel Better About Everything Else
2) Anticipate All Problems so You Will Be Prepared for Them
3) Apologize for Things That are NOT in Your Control
5) Try to Never Forget Anything
6) Take Imaginary Hints Too Seriously
7) Time is Money; Time is How You Spell L-O-V-E; Time is Precious; Time is Lost and Never Found Again
8) Resist Hard Things
9) When Everything Else in Your Life is Still Out of Control, Control Things That You Literally Have No Control Over
Fun and games aside, anxiety is a real thing and should be treated like one. If you think you have anxiety, get help!
PS. If you thought that last part was a little too serious, here is a picture of an adorable kitten.