Explanation of diauxie depended on the characterization of additional mutations affecting the lac genes other than those explained by the classical model.  Two other genes, cya and crp, subsequently were identified that mapped far from lac, and that, when mutated, result in a decreased level of expression in the presence of IPTG and even in strains of the bacterium lacking the repressor or operator.  The discovery of cAMP in E. coli led to the demonstration that mutants defective the cya gene but not the crp gene could be restored to full activity by the addition of cAMP to the medium.
The cya gene encodes adenylate cyclase, which produces cAMP.  In a cya mutant, the absence of cAMP makes the expression of the lacZYA genes more than ten times lower than normal.  Addition of cAMP corrects the low Lac expression characteristic of cya mutants.  The second gene, crp, encodes a protein called catabolite activator protein (CAP) or cAMP receptor protein (CRP).
However the lactose metabolism enzymes are made in small quantities in the presence of both glucose and lactose (sometimes called leaky expression) due to the fact that the RNAP can still sometimes bind and initiate transcription even in the absence of CAP. Leaky expression is necessary in order to allow for metabolism of some lactose after the glucose source is expended, but before lac expression is fully activated.
In summary:
* When lactose is absent then there is very little Lac enzyme production (the operator has Lac repressor bound to it).
* When lactose is present but a preferred carbon source (like glucose) is also present then a small amount of enzyme is produced (Lac repressor is not bound to the operator).
* When glucose is absent, CAP-cAMP binds to a specific DNA site upstream of the promoter and makes a direct protein-protein interaction with RNAP that facilitates the binding of RNAP to the promoter.
The delay between growth phases reflects the time needed to produce sufficient quantities of lactose-metabolizing enzymes.  First, the CAP regulatory protein has to assemble on the lac promoter, resulting in an increase in the production of lac mRNA. More available copies of the lac mRNA results in the production (see translation) of significantly more copies of LacZ (β-galactosidase, for lactose metabolism) and LacY (lactose permease to transport lactose into the cell).  After a delay needed to increase the level of the lactose metabolizing enzymes, the bacteria enter into a new rapid phase of cell growth.
Two puzzles of catabolite repression relate to how cAMP levels are coupled to the presence of glucose, and secondly, why the cells should even bother.  After lactose is cleaved it actually forms glucose and galactose (easily converted to glucose).  In metabolic terms, lactose is just as good a carbon and energy source as glucose.  The cAMP level is related not to intracellular glucose concentration but to the rate of glucose transport, which influences the activity of adenylate cyclase.  (In addition, glucose transport also leads to direct inhibition of the lactose permease.)  As to why E. coli works this way, one can only speculate.  All enteric bacteria ferment glucose, which suggests they encounter it frequently.  It is possible that a small difference in efficiency of transport or metabolism of glucose v. lactose makes it advantageous for cells to regulate the lac operon in this way.