The repressor is an allosteric protein, i.e. it can assume either one of two slightly different shapes, which are in equilibrium with each other.  In one form the repressor will bind to the operator DNA with high specificity, and in the other form it has lost its specificity.  According to the classical model of induction, binding of the inducer, either allolactose or IPTG, to the repressor affects the distribution of repressor between the two shapes.  Thus, repressor with inducer bound is stabilized in the non-DNA-binding conformation.  However, this simple model cannot be the whole story, because repressor is bound quite stably to DNA, yet it is released rapidly by addition of inducer.  Therefore, it seems clear that an inducer can also bind to the repressor when the repressor is already bound to DNA. It is still not entirely known what the exact mechanism of binding is.