Damage boosting allows you to progress through some areas more quickly. The only real requirement to it is that you have enough energy to survive the boost you are trying to perform.
By colliding with some kind of damage source (enemy, spike, electricity, et cetera), pressing the direction opposite to Samus's facing direction on the directional pad, and holding jump, you can initiate a damage boost.
The window with which to perform a damage boost is dependent on the damage source. Spikes and spike-like sources such as electricity provide 10 knockback frames, enemies provide 5 knockback frames, and enemy projectiles provide 4 knockback frames.
After the damage boost has been initiated, you can make it propel you even further by holding the direction you want to travel.
How successful your damage boost will be is determined by how you collide with the damage source and how fast you switch directions on the directional pad for the boost.
Arguably the first damage boost a runner should learn is the one on the way out of Kraid Eye Door Room, because it doesn't require a mid-air turnaround to save time.