I'm drawing largely on Bagnall's book on this. It's kinda an odd book; Bagnall was Chief of the General Staff for the British Army; so a military man by career, and only began a scholarly career after he retired. He worked at Oxford as a military historian and in addition to this book on the Punic Wars, he published on the Peloponnesian War. I don't find very much in this book by way of philological argument or source analysis; rather it's an explanation by a very good military man of the way the war seems to have been fought, if that makes any sense.

In the first place, Hannibal didn't cross the Alps with a giant army. He had about 26,000 men. That's not an insignificant force, but two years before against a Gallic invasion Rome had raised been able to field [see the discussion below for the distinction] 700,000 men - Romans and Italian allies. Those were men the land of Italy was able to support. Hannibal wasn't going to be increasing the burden of mouths Italy had to feed with this invasion. He only needed to find a way to locally source (as the kids these days say) his provisions.

You might think the bigger problem would be what Hannibal expected to do against Rome that could draw on a number of soldiers literally an order of magnitude larger than his, but these problems go together (along with the oft-asked question "why didn't Hannibal attack Rome directly?").

Hannibal's strategy seems to have been to attack Rome's treaty system, isolate Rome, and reach a negotiated surrender. The idea was that if Rome's army could be destroyed, the allies would defect to Hannibal, and he would supply (and indeed raise troops) from them. And in fact that is what happened to an extent; after three successive defeats culminating in Cannae, almost the entire Italian South went over to Hannibal. He was able to resupply and rearm. He wasn't able to entirely disrupt the Roman alliance system though. Rome retained enough allies to continue to put armies in the field and indeed to expand the war to Spain, the Balkans, and even Africa, while compressing Hannibal into a smaller and smaller portion of Southern Italy, until he was recalled to face Scipio at Zama.

So Hannibal never intended to rely on supplies from Carthage to fight his war; he was always intending to find a base of support in Italy, but he misjudged the resilience of Rome's alliance system.

Oh, and since Hannibal meant to reach a negotiated settlement with Rome by disrupting its political control of Italy, he never planned a direct attack and so didn't bring a siege train.