The background constraint controls the inertia you want to impose on the predictors (i.e. the smoothing in the predictor time series). It corresponds to an extra term in the WRFDA cost function.
It is defined through an integer number in the VARBC.in file. This number is related to a number of observations; the bigger the number, the more inertia constraint. If these numbers are set to zero, the predictors can evolve without any constraint.