Zerubbabel states the ostensible reason in the paragraph quoted in the OP:
You shall have no part with us in building a house to our God; but we
alone will build to the Lord, the God of Israel, as King Cyrus of
Persia has commanded us.
What lay behind Zerubbabel's "you vs. us" attitude here? The text, writing with hindsight, describes those who offered the help as "the adversaries of Judah and Benjamin." These were apparently remnant members of the northern tribes and/or non-Israelites who had immigrated from Assyria and intermarried with them during the Exile. They worshipped God and wished to help rebuild the Temple. Background is given in the Book of Kings.
And at the beginning of their dwelling there, they did not fear the
Lord; therefore the Lord sent lions among them, which killed some of
them. So the king of Assyria was told, “The nations which you have
carried away and placed in the cities of Samar′ia do not know the law
of the god of the land; therefore he has sent lions among them, and
behold, they are killing them, because they do not know the law of the
god of the land.” Then the king of Assyria commanded, “Send there one
of the priests whom you carried away thence; and let him go and dwell
there, and teach them the law of the god of the land.” (2 Kings
17:25-27)
The intermarriage issue was not only a problem for the northern tribes, but also for the Judeans:
all the men of Judah and Benjamin assembled at Jerusalem within the
three days; it was the ninth month, on the twentieth day of the month.
And all the people sat in the open square before the house of God,
trembling because of this matter and because of the heavy rain. 10 And
Ezra the priest stood up and said to them, “You have trespassed and
married foreign women, and so increased the guilt of Israel. 11 Now
then make confession to the Lord the God of your fathers, and do his
will; separate yourselves from the peoples of the land and from the
foreign wives.” (Ezra 10:9-11)
Thus we can deduce several factors in Zeruabbel's decision to exclude those who offered help.
He may have had instructions from Cyrus not to cooperate with them.
He did not want to share credit for rebuilding with those already living in the area, especially those of impure bloodlines.
He may have sensed that they were already "adversaries" in some sense.
Eventually, those he excluded undermined Zerubbabel's authority after their offer was rejected. Later, they would build their own temple on Mount Gerizim and be known as Samaritans. But were they actually adversaries when they offered aid, or did they develop the adversarial posture mentioned in the text only after they were rejected?
