It is a combination of geography and ideology.

Egypt is surrounded by natural barriers that make it difficult for an army to attack them, such as the Western and Eastern deserts, the Ethiopian highlands to the south and the Mediterranean Sea to the north. The easiest way to get in to Egypt was across the Sinai desert, or by sea. As such, as long as Egypt controlled these entry points, it was difficult for outside forces to invade with large numbers of troops. So they were not seriously threatened.

Egypt has similarities with modern nation-states, in that there was generally one ruler governing a territory. But in this, it is fairly rare. The norm in early periods was the city-state, and would remain so until empires came along with the Akkadians in the 24th century BCE. Even these empires are better understood as centralised control over various culturally distinct city-states. But it really isn't until the Neo-Assyrians that we get a really successful empire, most previous attempts lasting only a couple of hundred years (eg. Akkadians; 3rd Dynasty of Ur; Old Babylonian, LBA empires like Mitanni or the Hittites). The reason for this is similar, geography. Mesopotamian cities lacked the same natural barriers that Egypt enjoyed. For example, Babylon was sacked in 1595 BCE by the Hittites, who basically wandered down from Anatolia for the ancient equivalent of a 'daytrip'. These lack of natural protection and enemies on all sides made Mesopotamian empires short-lived (Van De Mieroop, A history of the ancient Near East).

So despite the civilisation or culture that Mesopotamian city-states shared in terms of religion, language and other traditions, there is a tendency to keep this independence. Those that did expand typically overextended and were unable to control the territory they had obtained (See Yoffee, Myths of the Archaic State).

Also important is ideology. Egypt as a single state is a fairly simplistic view. Egypt was really two states, Upper and Lower, with a strong ideological glue holding it together. The slightest pressure and it would crack and split into two halves again, as demonstrated in the various intermediate periods. So additionally, ideological structures that united the territory, such as the double crown representing both Upper and Lower regions. Those structures in Mesopotamia were not as successful, at least at first. The Mesopotamian king was largely an administrator, whereas the Egyptian king was a god. It is far more likely a god is going to hold a territory together than an administrator, even if he communicates with the gods (Frankfort, Kingship and the Gods).

Mesopotamia ruled in ways other than divine right, but in more of a corporate style, like the Roman Empire did later on. Each city had its own ruler (and individual god/goddess) and as long as a city paid tribute to whichever empire was in control at the time (there were a few different ones at different periods, and your question didn't specify which empire you were wondering about), then the empire would provide protection from outsiders/hostiles. So, whilst Egyptian rules ruled by divine right, the Mesopotamian rulers ruled by military (essentially).