Like already mentioned, ZeroG is meant to have the flair of the "Cannonball" movies or games like Midnight Club.
A bunch of speed fanatic maniacs from all around the galaxy meet every year for the ZeroG Grand Prix. Everything that flies is allowed, as long as it fits into the lose rules of the race. The winner of the Grand Prix will earn a lot of money from sponsors. But the most desired thing amonst the field is the honor to be a ZeroG winner. Of course the races are prohibited by the police and therefore they try everything to stop the madness either by hindering the race or by preventing the entire Grand Prix. Because of that most of the races take place in areas where nobody will suspect them. This is another kick for the racers. Sometimes the races are simply dangerous because a lot of other vehicles are there, too (e.g. Neo Manhattan) or because the race takes place in a warzone (Raid over Serathin 4). The locations are very diversive and take everything a racer can give. To prove your skills is what makes the race.
The winner of the Grand Prix is of course a very respected person among the pilots and it's an extraordinary honor to have won a ZeroG Grand Prix.
The player (or the players in multiplayer) are contestants of this race. To win, everything goes, even the useage of weapons. Everybody tries to shoot down the others and not getting shot and win the Grand Prix.

To put it short:

It's all about speed, cool vehicles, money, hot chicks and respect