OED is a bit pricey, but one can always use etymonline for a degree of insight. It's free, solid, and iirc uses OED as one of its main sources.
As for the meaning shift, here's a rough-and-ready story one could piece together:
You start with the verb for using a sieve. This naturally lends itself to the meaning of discerning, separating, distinguishing - because that's what you implicitly do when you use a sieve.
From there you get: to be of importance to oneself, affect one's interest, and also worry - as these are the states of mind involved.
This in turn leads to the meaning of being busy, engaged in a task, focused on doing something - from the more superficial appearance of the former sense.
Along the way the verbs are getting nouned, as is the wont of the English language.
From the meaning of being important to oneself emerges a noun for regard, reference.
The noun and the verb keep cross-pollinating as they evolve, though. The regard, reference meaning largely fades away, while from being busy etc. we get the noun meaning for business, establishment, occupation.
And that would be the end of the story. However, I strongly suspect that meaning itself faded away from English, and the word's modern use in this sense is largely influenced by the German konzern, which is a large conglomerate of companies common therein and in other parts of Europe. Where, having done some business with the Germans at some point, the English language dusted off the old word with similar enough sense and called it a day.
As for why would concern even develop when business was already available - probably the same reason beef and pork developed despite cow and pig already grazing the linguistic landscape. It's them Normans putting their ill-fitting Frenchy hat onto the local Germanic bumpkins. So that raising cows may be the business of the common folk, while the fancy aristocrats concern themselves with eating beef.
