How might gut bacteria be important for immune responses to a lung infection? When influenza virus infects the lung, development of immune defenses depend upon a complex of several proteins called the inflammasome. This structure is needed for the production of cytokines that promote adaptive immune defenses: antibodies and T cells. These cytokines are also needed for the activity of dendritic cells, sentinels that sense a virus infection, and travel to the nearby lymph nodes to inform T cells that there is a problem.
If you are wondering about the practical consequences of these findings, read the last paragraph of the paper:
Because antibiotic use is prevalent in the treatment of respiratory infections, our results imply a possible deleterious effect of such treatment in initiating proper immune responses to influenza virus. Conversely, it will be important to determine whether probiotic therapy can be explored for immune-stimulating effects during the flu season.