The vast majority of the unoccupied systems are unclaimable space - they're owned by non-player (NPC) factions and are effectively a 'neutral zone' that any player (with a few exceptions) can visit. The center area is all NPC space and there are a few patches of it in other places.
The lines on the map represent stargates which link between systems. systems are organised into constellations, constellations into regions. blue lines are intra-constallation, red are inter-constallation, and magenta are inter-region (and often the longest ones).