They're not as exceptional in the Ancient Near East as popular media might make them seem; a lot of the wartime atrocities they carried out that are reported in sources like the Hebrew Bible existed elsewhere in the ANE, though you could argue that the Neo-Assyrians often carried them out on a larger scale.

The history of ancient Assyrian urban states is usually divided up into the Old Assyrian, Middle Assyrian, and Neo-Assyrian periods. The Old Assyrian states are notable for their trade colonies in Anatolia and the importance of their merchants. Assyria was subjugated by the kings of Akkad, and when they fell the kingdom spent another couple of centuries before engaging on campaigns of conquest beginning in the reign of Shamshi-Adad I (19th-18th centuries BCE), conquering the then-powerful state of Mari which was centred in Syria and extending its power into Anatolia. Soon after this, however, Assyria was subjugated by Hammurabi's dynasty of Babylon. The centuries after this are quite poorly known, but when the Mitanni entered Mesopotamia and Syria they seem to have made the Assyrian kings their tributaries from the 15th-14th centuries BCE until Assyria successfully rebelled and restored their own independence.

Having broken the power of the Mitanni, the new Middle Assyrian empire was able to seize much of their former territory and more. King Tiglath-Pileser I (12th-11th centuries BCE) was able to reach the Mediterranean and subjugate many of the cities of Phoenicia. The Middle Assyrian Empire waned a little after this high watermark but is considered to have transitioned into a new phase, the Neo-Assyrian Empire, with the accession of Adad-nirari II. Exacavations of cities like Nineveh and Nimrud in the 19th century, with their impressive bas-reliefs, monumental sculpture, and cuneiform libraries, triggered the founding of the field of Assyriology in Europe. These are also the Assyrians who appear in the Hebrew Bible, such as Sennacherib seizing the cities of Judah in Isaiah 36 and Tiglath-Pileser III in 2 Kings. Their role in destroying cities and deporting populations as the rod of God's wrath left them with a violent and terrifying legacy in the Abrahamic imagination.

None of these things are really unique to the Assyrians, though. The Egyptians, Hittites, and Babylonians also engaged in mass deportation policies. Kings such as Šulgi of the Ur III dynasty in the 21st century BCE proclaimed their destruction of city after city, often using them as year names (since years are often named retroactively according to an important event of that year) and used these conquests for their own propaganda. Titles that the Assyrians used to universalise their own claim to empire such as Šarru Kibrat 'Arbaim (King of the Four Corners of the World) or Šarru kiššat māti (King of the Universe) are much older than Neo-Assyria, dating back to the kings of Agade in the 3rd millennium BCE. There was a religious element to their conquests and imperialistic ideology, but this had been true for every imperial state in Mesopotamia; Sargon claimed to conquer the Upper Euphrates by the will of the god Dagan, for example. In the religion of the time, any conquest was necessarily the result of the God's favour. The proof could be found in the victory.

There are two things that might make the conquests and suffering inflicted by the Neo-Assyrians stand out in the ANE: their scale, since the Neo-Assyrian empire at its peak did have a broader territorial extent than any Ancient Near Eastern state before it, and the way that these deeds have been remembered both in the propaganda spread by their kings and in the annals of the powers that they fought.

The god Aššur is actually attested from all the way back in the reign of Ušpia, a king who reigned in the 21st century BCE. The development of the god is still quite mysterious, though W. G. Lambert wrote a useful paper on the topic (The God Aššur, jstor). From the earliest attestations, he is tied to one particular rocky outcrop at Qal’at Sherqat in Iraq which would develop into his namesake city and house his temple, the Ešarra, which continued to serve as his temple until the end of the Neo-Assyrian period. As Assyria developed into a major player in the Ancient Near East, the god's close association with royal power caused his profile to rise higher and higher. However, there's very little evidence of any attempts to integrate him into the families of the gods of Babylonia, which the Assyrians also venerated, until the reign of Sennacherib (704-681 BCE). To some degree the Assyrians attributed qualities of the major god Enlil from Southern Mesopotamia to Aššur, co-opting his status, but he is also written into stories about Marduk, the chief god of Babylon, and Anšar, a primordial figure from Babylonian cosmogeny. What is always central, though, is his sovereignty, initially over the city and kingdom but eventually over the entire universe as he becomes the pre-eminent deity and universal king. In Neo-Assyrian ideology, even the 'King of the Four Corners of the World' is merely the governor ruling the world on Aššur's behalf.

It's difficult to attribute Neo-Assyrian militarism, such as it was, to a religious transformation. The Assyrians picked up military innovations and imperial social technologies from the the kings of Agade, the Mitanni, and the Babylonians, who were certainly not averse to war and destruction. As far back as Agade and Ur III there are kings who seem to campaign every year and speak with pride of the treasures and captives they brought back with them. Their religion legitimises conquest and the gods and temples have a mutually beneficial relationship with a successful conquering king, but the religion seems to be transforming in response to empire (whether Assyria was building or receiving it) rather than the other way around.

If you'd like some recommendations for reading material, I can wholeheartedly recommend van de Mieroop's A History of the Ancient Near East, ca 3000-323 BCE as a general summary of the history of the region in that time, and Karen Radner's Ancient Assyria: A Very Short Introduction as an excellent and very readable brief overview of Assyria. It manages to be immensely more accessible than any other general work on the Ancient Assyrians.