Dear HSAA,
With a brief check, I could not actually locate the origin of this term... However, I may have a reasonable guess!
Think of a safecracker...listening, feeling, fingernails filed to the quick...and finally, AHA, success! He (or she) senses those tumblers fall into place with a wonderfully satisfying ... click!
So, click as slang can have your meaning, (we just really click, you know?) or it can mean "to be come understood," (I had to read that chapter 3 times before it finally clicked)...
But like the safecracker's tumblers, something just falls into place.
http://onlineslangdictionary.com/meaning-definition-of/click* * *
Here is something interesting I learned...in Canada, click can also refer to a kilometer! As, "Ottawa to Toronto is about 450 clicks."