Since this is likely to be the only time you ever use the word "twiddling" or "twiddle" people aren't really sure how to say it. Instead you hear "twittling" and I even heard someone today say, "twittering". But the actual word is "twiddling".
You twiddle your thumbs when you are bored.
There isn't a clear etymology to this phrase.
Another thumb/ boredom idiom is "standing around with you thumb up your ass"
I did a great deal of research into the origins of this one with no luck. I'm going to assume it's probably because whoever did come up with this isn't too keen on getting credit. I would assume alcohol was somehow involved.