The great thing about Big Rigs is that it lets you make up your own storyline. To me this adds to the excitement. Other games like Final Fantasy force the storyline on you, often making you play as a non-heterosexual persongot punk with a sword who's trying to stop some other non-heterosexual persongot. Gay and boring. Anyway, here is the storyline I like to imagine as I play Big Rigs:

The billionaire tycoon of the world's most successful grocery store is trying to set a new standard for transportation efficiency. In an attempt to find the world's greatest truck driver, he has staged a multinational competition. The winner gets the esteemed job of transporting groceries and supplies for the world's most successful business. It is a truck driver's dream. Thousands of the world's best truck drivers entered the competition, and now only you and one other driver remain. Whoever wins this race gets the job of a lifetime, the title of "World's Greatest Truck Driver," and international fame.

Sadly, as soon as the race begins, your rival has a heart attack and dies. You mourn the loss of this great truck driver momentarily; although this makes things easier for you, he was a great friend and a wonderful truck driver, and you wouldn't have ever wished this upon him. But despite a devastating loss, you're forced to regain concentration. There is still a very challenging task ahead.

The track in front of you is legendary. It was painstakingly engineered to be the greatest test of a truck driver's endurance, perseverance, and psychological durability. Several of this track's challenges have you particularly worried. There is the infamous police roadblock, which has been known to devastate even the most ambitious of truck drivers. There is the Grey Void, whose seducing gaze has steered dozens of drivers off track and swallowed them whole. There are fake bridges and elusive buildings designed only to confuse you and test your psychological ability under stress. You are not afraid to admit that you are scared. But you know the finish line is within your reach, and you are prepared for what lies ahead.

I shall now give you +1 CP every time I log in for eternity.
 

This does not explain the illegal cargo though... 

You didn't wait to let BIA explain the part of the story where it's revealed that the Grocery Store is just a front for a drug-smuggling operation, and since the rig runs over the Grocery Store owner and the soul of the evil HWSNBN floats out of the body screaming "CURSES!  I MUST FIND A NEW HOST!"
And I think we all know who that was...

What a great story. And the great thing is we can all have our own unique versions of the story, giving BROTRR massive replay value due to the unlimited amount of storylines.

This does not explain the illegal cargo though... 

I believe the groceries are being transported to an evil Communist regime where imported groceries are illegal. However Grey Void has presented a viable theory as well. And truly anything is possible, as WhySoWinner pointed out.

Nice read. Would read again.