How was the Achaemenid Persian Empire able to conquer so much territory in the 6th century?

Military force.

Were the surrounding powers weak?

Not necessarily. So, first, you need to understand a couple of things:

1. It was not the "Achaemenids" who conquered the territory, but rather the Person-Elamite Kingdom of Anshan. Only with the coup of Darius I (the first actual Achaemenid) around 522 do we get the assertion of a clearly "Persian" identity.

2. Cyrus' conquests, as well as the state and precise extent of his kingdom, are poorly understood.

From Babylonian records, we know that his conquests started with subduing the Medes, who controlled what amounts to Northwestern Iran and probably part of the Transcaucasus. We don't really understand the political organization of the Medes - they were clearly able to marshall considerable military force, but they seem to have been something closer to a tribal confederation than a "kingdom" proper. The Greek record is bloated with fanciful tales concerning the relationship between Cyrus and the elderly Median chieftain or king Astyages, which makes it difficult to evaluate. But I'd say general impression of the Medes is that they were one of those recurring groups of seminomadic tribal peoples who raided the borderlands of neighbouring empires, perhaps serving as mercenaries in times of need.

In any case, having subdued the Medes and presumably integrating Median troops into his army, Cyrus seems to have turned his attention to Asia Minor, probably settling the war with the eventual taking the fort of Sardis. He may have engaged in campaigns in Central Asia for some time, before encroaching on Babylonia, sacking its cities and taking the city of Babylon in 539.

One of the traditional explanations - that owes much to Persian propaganda - is that Nabonidus, the last Babylonian king, had elevated the moon god Sín above Marduk, angering the Marduk priesthood. He apparently spent the decade between 553 and 543 in the city of Tayma (in the northwest of what is today Saudi Arabia), leaving the governing of Babylon to his son Belshazzar. The problem is that this information comes from accounts that were put together after the conquests of Cyrus, and as such, were written to legitimize his rule. Much has been made of conflicts between the Marduk priesthoods and Nabonidus that Cyrus exploited, but this is in my opinion rather fanciful.

At any rate, Nabonidus had seized power in a coup against the Chaldean dynasty (the dynasty of Nebuchadnezzar II) and its king, who was a child at the point of the conspiracy. Nabonidus' own background is unclear, and it may well be that he suffered legitimacy problems that Cyrus exploited. But there is little basis for anything except speculation, given our poor understanding of pretty much every involved figure.