To "veer off" means to turn slightly.
Telling someone to "veer off to the left" is different from telling them to "turn" to the left. "Veering off" means turning at an angle of about 45 degrees. In a car, a "turn" usually requires you to stop, but "veering" doesn't.
The word "veer" can be used in other situations as well. It's not just for cars. Storms can also "veer off":
They were expecting the hurricane to hit the South Carolina coast, but it veered off and didn't hit until Virginia.
When someone drives their car off of the road accidentally, we say that they "veered off the road":
Barry fell asleep at the wheel, veered off the road, and crashed into a telephone pole.
Since "veering off" means that you're not going straight in one direction, people use it to describe things that are not happening in the way that was expected, like a conversation:
The conversation suddenly veered off into the topic of birth control.