It's a really good question, and rectus abdominis (abs) was a tough one for me to figure out personally. For a lot of exercises like the deadlift and front squat (and back squat, but so much more clearly on the front) you want to have your back locked into a neutral position.
For a demonstration, try locking your forearm into a 45 degree bend at the elbow (halfway between the classic bicep muscle squeeze and being straight armed). You'll notice that your tricep and bicep are both tensed: the cool term for this is that one is a stabilizer and the other is an antagonistic stabilizer.
In order to lock your back into a neutral and safe position, so that the muscles are taking the load and not your vertebrae, you need both your back muscles and your abdominal muscles locked, just like your bicep and tricep.
Specifically answering your question of What is the proper use of the abdominal muscle to reduce injury?, I'll keep stealing from exrx with this quote regarding abdominal weakness:
Increased risk of lower back injury can occur during hip flexion,
extension, stabilization and back extension activities. Erector Spinae
muscles can hyperextend lower back more than usual if abdominal
muscles are weak. The abdominal muscles tilt the pelvis forward,
improving the mechanical positioning of the Erector Spinae,
specifically when the lumbar spine becomes straight. When abdominal
strength/endurance is not adequate to counter the pull of the
antagonist Erector Spinae under load, these low back muscles are put
at a mechanical disadvantage (active insufficiency) further placing
additional stresses on these very same lower back muscles. Iliopsoas
can pull on the spine during hip flexor activities if the abdominal
muscles are weak. Risk is compounded when abdominal weakness is
combined with hip flexor inflexibility.
As a reminder to myself, when doing a lift where I need my abs tight, I try to imagine one of my children jumping onto my stomach. Or maybe the split second before you know you're about to be sucker punched: that tensed up feeling where you harden your abs and increase abdominal pressure. This is also linked closely with the valsalva maneuver.
Some people frown on holding your breath when you lift heavy weights, but in my experience the protection to your back is worth it and it's completely natural: if you were to try to push a car to roll-start it (as an example), you would probably hold your breath on your first push.