When and what lead jews to leave Israel after the Roman conquest and spread out through out the world?
Actually, the Jewish diaspora (i.e. their spread outside Israel) began long before, with the Assyrian conquest of the Jewish Kingdom and the deportation of the Jews to other territories, a common practice amongst the Assyrians, and with the Babylonian Captivity. Later, the Persians conquered Babylon and allowed the Jews to return to Israel, but many just didn't. The Greeks and the Romans would later conquer the Eastern Mediterranean.
Broadly speaking, at first, the Romans were more or less lenient conquerors, allowing the Jews to continue their worship and have their own rulers as long as they acknowledged Roman overlordship. After the First Jewish-Roman War and the uprisings that followed, the Temple of Jerusalem was destroyed, the Jewish leadership was executed and many Jews were exiled, and with that the center of political and religious power shifted from Jerusalem and the Temple to the local Jewish communities throughout the Empire.