How should I keep the vegetables from flying off the board when I cut them?
Use a sharp knife, dull knives don't slice, they split the same way an axe splits logs and that will generate that sideways force that throws carrot bits.

Also learn to practice your slicing and chopping technique. You shouldn't be dropping the blade straight down like a axe or guillotine. You should be moving the blade in a orbital movement, so that you are slicing in a forward direction while the blade is on its way down to and through the carrot. By slicing properly much less of your veg will have that energy buildup because they won't "snap" apart but will be sliced apart neatly.

Search YouTube for Basic Knife Skills and watch several videos, then practice, and start slowly. You will get faster with time and time only, don't rush it or you'll get sloppy and may hurt yourself.