Allolactose is called an inducer because it turns on, or induces the expression of, the lac genes.
The presence of lactose (and thus allolactose) determines whether or not the Lac repressor is bound to the operator.
Allolactose binds to an allosteric site on the repressor protein causing a conformational change. As a result of this change, the repressor can no longer bind to the operator region and falls off. RNA polymerase can then bind to the promoter and transcribe the lac genes.