Recall that the Second Punic War started after Hannibal attacked the city of Saguntum. Saguntum was a city on the eastern coast of Spain, near modern-day Valencia. It was well within what the Carthaginians considered to be their sphere of influence. In fact, it was a mere 150 miles from New Carthage, the main Carthaginian stronghold in Spain. Saguntum had perhaps 1/5th the population of Rome.

It took Hannibal EIGHT MONTHS to capture Saguntum!

Taking Rome by siege would have been an impossible task for Hannibal. His army was smaller than at Saguntum; he had no support from his home base and no way to reliably get rations for his troops; he would be in hostile territory, with cities allied to Rome and possibly a Roman army in his back (if the army sent to Spain would be recalled) - all of that for months, possibly years. There was no way Hannibal could've had a successful siege of Rome.

What then was Hannibal's goal?

In Hannibal's mind the cities allied to Rome were only allies out of fear. They were forced or pressured into the alliances, and would likely throw off the Roman yoke if they could rally behind someone capable of fending Rome off. This is what Hannibal intended to do. Break the Roman Empire by alienating its allies and reducing it to just another city-state.

Unfortunately for Hannibal, he had misread the situation. By the start of the Second Punic War, some of Rome's allies had been allied to Rome for a century or more. While the allies were clearly subordinate in the alliance, many had found that there were benefits to the alliance. So far fewer took Hannibal's side when he was marching from north to south through the peninsula. After the Battle of Cannae in 216 B.C. some cities in the south, most prominently Capua, went over the Hannibal. A consequence of this was that Hannibal could not continue marching freely across Italy, as those cities expected protection.

With the help of its loyal allies around the city, Rome made a steady recovery. By 211 B.C. Rome had recaptured Capua and made an example of it. Much of the population was killed or sold in slavery, most of its lands became public lands belonging to Rome. No other allies defected to Hannibal after that.

It is here that I want to address your statement that "He defeated Roman armies left and right and managed to plunder Italy for 15 years straight." The defeating of Roman armies "left and right" happened in the first three years of the war. After that there were very few major engagements between Rome and Hannibal. Also, the "plunder Italy for 15 years straight" has to be qualified. After those first three years Hannibal was pretty much contained to southern Italy, and the Romans were content to keep him there. While Hannibal was achieving very little in Italy, Rome first threw the Carthaginians out of Spain and then invaded their homeland in northern Africa.