There are 3 ways to understand what the "red line" means in hockey.
First, it is the line where the opponents meet as equals before the puck drops. It is the spot where neither team can claim a territorial advantage.
Second, it can be understood as the physical demarcation between "us" and "them" dividing hockey's neutral zone (the hockey term for the space between the blue lines). Throughout hockey's history, progress across the red line used to be restricted. Prior to the 2004-05 NHL lockout, players could not make a pass across two lines without play being whistled down. Following the lockout, in an attempt to create more offence for both teams, the red line was effectively removed and two-line passes legalized though the red line remained intact. However, the line itself remains painted on every arena in the world. Therefore, the physical line itself is a sort of hockey artifact from a time when the game was more restricted.
Third, outside of the hockey context, "crossing the red line" connotes transgressing a point of no return after which serious, irreparable repercussions are possible. In many ways, this transgression represents the establishment of the Indian residential school system that created the Black Hawks.