If you grow E Coli in a petri dish containing the sugar glucose, its happy. Its got enzymes that chew glucose and extract energy. But Jacob and Monod knew that if you take the glucose away from E Coli and give it the milk sugar lactose instead, it will continue to grow. First, however, it needs to express a new set of protein enzymes that enable it to munch lactose. As it turns out, these proteins are all coded for by a single gene, the lac gene. (Note that this single gene has multiple gene products. This situation is much more common in prokaryotes, and such a gene is called a polycistronic gene.)
Now, if E Coli had this gene on all the time, it would make lactose-eating enzymes even when there wasnt any lactose arounda waste of energy and materials. So it must have a way to recognize that its rich in lactose and needs to transcribe the lactose-munching genes.