One solution, historically often employed in times of dire need, is to push into service inexperienced soldiers-in-training and even young men. Even though they may not be as efficacious, per person, as a fully trained soldier, they are still better than nothing.
The war, of course, is an analogy of what happens every time our bodies meet a foreign pathogen - a vicious immunological battle breaks out between ourselves and the invader. In this context, more white blood cells (predominantly neutrophils) are required, and the bone marrow starts hyper-producing them, pushing out the additional supply as soon as it is ready. However, a further option is to dispatch slightly immature forms of the neutrophils - neutrophils-in-training if you will - on to the frontline.
This is what "left shift" refers to (at least in the context of medicine, anyway). The immature neutrophil forms tend to have less lobularity to the nuclei; their nucleus tends to be shaped like a curved band, hence 'band cells'.
As far as I know, the odd name comes from the fact that textbooks traditionally showed the haematopoeitic* series from left (most immature) to right (mature cell) - and so a 'left shift' would equate to an increase in the proportion of immature cells in the peripheral blood.
* blood cell development