When $E$. coli is exposed to both lactose and glucose, the organism first metabolise glucose and then temporarily cease to grow until the genes of lac operon become included to provide the ability to metabolise lactose. Allolactose is called an inducer because in its presence it turns on, or induces the expression of the lac genes.
Related Theory
When lactose is present outside the cell. it crosses the cell membrane and acts as an inducer of the operon. It does so once lactose is broken down to create allolactose. The lac operon is then membrane facilitating lactose transport into the cell, and \& beta galactosidase, which eats up lactose to Make glucose molecules. And beta-galactosidase also makes allolactose. This leads to a positive feedback loop.