Ugh, I accidentally deleted a fairly long post contextualizing this, so this will be a bit briefer:

It is a little unclear, which is actually fairly common in Mesopotamian history because narrative historical data tends to come from highly centralized institutions, such as temple archives and royal records. Breakdowns in those institutions tends to lead to similar breakdowns in historical narrative, and so Mesopotamian history tends to be characterized by periods of near absurd levels of detail punctuated by periods of complete fog. As an example, the Neo-Assyrian Empire went from an absolute pinnacle of strength, the destruction of all of its traditional enemies such as Urartu and Elam, to being conquered by a coalition of enemies in about twenty years. Our source for this conquest comes from Bablyonian records, which are not terribly interested in the internal Assyrian explanations. There is a similar dynamic at play with the Neo-Babylonian Empire: the conquest by Cyrus the Great was about twenty years after the death of the famous Nebuchadnezzar II.

This is exacerbated in this instance because our records on Cyrus himself are painfully short. There is the famous (and much misinterpreted) Cyrus cylinder, in which Cyrus gave his own version of events: he conquered because he had the favor of the gods, because the king Nabonidus was wicked. There may be an element of truth to this, in that Nabonidus took as his personal tutelary deity the moon god Sin which Cyrus used to claim that he neglected the traditional worship of the gods of the city, chiefly Marduk. Is this accurate? Hard to say. It has some support in the maddeningly terse Babylonian temple chronicles, which has a lot of statements like this:

Tenth year (546/545): The king stayed in Temâ; the crown prince, his officials and his army were in Akkad. The king did not come to Babylon for the ceremonies of the month of Nisannu; Nabû did not come to Babylon, Bêl did not go out of Esagila in procession, the festival of the New Year was omitted. But the offerings within the temples Esagila and Ezida for the gods of Babylon and Borsippa were given according to the complete ritual.

This is often used to say that Nabonidus alienated the people of his empire and the temples by suppressing traditional worship in a fit of religious extremism, but that may be seeing a bit of a mountain in a molehill. Regardless, the traditional explanation goes something like this: Nabonidus was a bad and unpopular ruler, which was exploited by Cyrus, who led a renewed Elamite/Persian/Medean army to a conquest that was generally pretty popular, and there was much rejoicing. This view has the advantage of being broadly in agreement with the Greek historian Herodotus' account (although I think Herodotus account is so confused that it isn't terribly helpful).

I think, however, that there is a somewhat more interesting a deeper explanation as well. Mesopotamian empires, for all their flair, tend to be pretty shaky. Under a strong king such as the Assyrian king Ashurbanipal they could project power highly effectively, but small cracks tended to bring the foundations down quite quickly. We can see this in marvelous detail with the conquest of the Persian Empire by Alexander--despite its administrative arms still being perfectly functional it collapsed in only a handful of years of pressure. Much of the reason for this is because regions within the empire had a high degree of autonomy and independent power. Their loyalty to the Persian center was conditional--when something else showed up they had no problems defecting. I suspect that Cyrus' conquest of Babylon was quite similar to Alexander's conquest of Persia in this respect.