They had only declared statehood a very short time ago (like I think a few days beforehand) when they were attacked by almost all of their neighbours, and their huge armies? How were they able to actually win the war?
Prior to WW II the British were trying to reconcile their promises to various parties over the Palestine issue which was turning into an ulcerous albatross around their neck. By 1937 the Peel Commission returned a report that the Palestinian Mandate was unworkable and recommended a three way partition between a Jewish and Arab state, and a British enclave. The Arabs were miffed at this proposal that would see them lose a considerable chunk of land and rioting ensued that eventually developed into a full revolt which would be suppressed in 1938. So on the eve of WW II, the Palestinian Arabs had been disarmed and most of their senior leadership were in exile. There was fighting amongst themselves between moderates and more radical factions. Complicating things further was other Arab states interfering to serve their own personal goals.
In contrast, the Palestinian Jews saw Britain as their ally and many had served as auxliaries in the Arab Revolt. So their existing (and illegal) milita, the Haganah, was intact and many members had gained valuable military experience. Come the war, the British recruited enough Jews to form a brigade force. This meant after the war, they returned bringing even more relevant military experience to the fledgling Israeli army. Note the British systematically recruit Arab soldiers like they did with the Jews.
As for the war itself, the various Arab states simply weren't as well trained as the Israelis. Their was no central high command and each nation did it's own thing. Jordan had the best equipped and trained force but was concerned with keeping it's hold on the West Bank.