I've answered this question before here. There were multiple reasons for him to head East - historical, cultural, political and financial. Both the ancestral enemy of the Greeks and Macedonians and the greatest riches were in the East; there was nothing much in Italy that was worth having, and no way to justify claims to it.

Once this campaign was under way, Alexander's aim in going ever further east was his attempt to reach the outer river (Okeanos) that the Greeks throught surrounded the world, so that his empire would have no eastern border. However, it turned out that Asia was a good deal larger than the Greeks knew, and he eventually turned back and began making plans to invade Arabia and Africa instead.