What it does
The extension is how your network gets into O. It brings your connections across from LinkedIn, and keeps doing it quietly as you browse rather than asking you to run an import.
Open someone's profile and the extension notices. From there you can pull their contact details into O, see what they are about, and keep the record without copying anything by hand.
Why it matters
Most of what you know about your network is already sitting in places you visit every week. The extension is the bridge between those places and a memory that holds them together, so the network you already have becomes a network O can actually work with.
It is the first thing to install, and after that it mostly stays out of the way.