Luminosity is necessary in order to turn number of interactions to crossections, because theories provide crossections to compare with experiments.
Experiments measure number of interactions. A well known crossection, as is Bhabha scattering, substituted on the right will give the luminosity to be used in the other observed interactions in the experiment. Of course it works for electron colliders, but the logic of calibration is the same for other colliders too.
From the formula one sees that the higher the luminosity the more number of interactions per unit time. As the statistical error on the values extracted from the data come from the number N, the larger the number the better the accuracy of all measurements.