Basically, running forwards and strafing left or right at the same time results in you moving diagonally roughly 40% faster than just running forwards alone. It's a vector sum kind of thing.
It's standard to speed-strafe throughout the whole of any given speed run, except when it's necessary to look or shoot at something directly ahead of you. It becomes very hard to unlearn this practice for games where it doesn't work, like TimeSplitters 2.
Switching strafe directions while moving continuously forwards, for example at a corner, results in your strafing speed (the left-to-right component of your motion) dropping from 100% in one direction all the way to zero to 100% in the other direction. During this time, your horizontal velocity is NOT at its maximum which means that for the split second it takes you to fully change direction - even if it takes you no time at all to move your thumb - you are NOT moving at top speed. Therefore, it is good general practice to switch strafes as little as possible, especially on flat speed levels like War. When you do switch strafes, try to arrange to do so at places like doors, where you usually have to slow down fractionally for the door to open anyway.
Speed strafing also cultivates the highly advisable practice of strafing around potentially dangerous corners instead of blindly walking out into gunfire, and allows the activation of wall-mounted buttons and objects while moving at top speed parallel to the wall, which is rather handy.