As I understand it, it's a sort-of codeshare. So ASW flights will still be sold on their own website, but also promoted under T3 (Eastern) flight numbers via Eastern's links with the travel agents systems (GDS systems). So it gives ASW additional exposure in markets outside the South West where perhaps they aren't widely known at present.
The "extra machine" is, I believe, a Dash 8-200 due to arrive in the summer to act as a backup aircraft replacing the J41 they have on lease from Eastern at the moment. I guess the idea is it save money to have their own backup rather than an Eastern unit, and means their own crew can fly it, rather than Eastern's.