The NFL especially has been the target of numerous anti-trust cases over the years. It dates back to two main events, the Sports Broadcasting Act of 1961 and the NFL-AFL merger.
At the onset of TV the NFL began league using league wide tv contracts instead of allowing seperate ones for each team (like FSN has with the Penguins/Pirates). The courts saw this as an illegal act under anti trust laws. The SBA of 61 granted an exception to this, and it allowed the teams to sell packages to the networks (such as CBS broadcasting AFC, Fox NFC, NBC Sunday Night, and ABC/ESPN MNF). The NFL's "blackout" rules were part of this because the league broadcasting Team A's game on TV could be construed as competing against Team A's ticket sales.
The merger was seen as losing competition in the professional football market. The leagues wanted it seen as two entertainment corporations merging together (like Marvel Ent and Disney's recent merger). They wanted to be recognized as just another company looking for your entertainment dollars. To legalize the merger they were granted an anti-trust exemption which meant that they could not be sued for being a monopoly unless they specifically tried to stop another league from forming, or were using their status to otherwise reduce competition.
Opponents will say that the league is basically 32 businesses acting in collusion together. They work together to decide the number of teams (thus barring "new" businesses/teams from the market), access to stadiums, access to tv, access to players etc.
If the Supreme Court rules in the NFL's favor, they believe it (and by extension the other pro leagues) will forever be viewed as single entity entertainment corporations. It will allow the leagues a lot more leeway in decisionmaking, because they will not have to worry about upsetting anti-trust lawsuits, whether from TV networks, apparel venders, or unions. They will have an easier time setting salaries, making licensing deals, and whatever else they choose.