Mostly, the link to the new moon appears in one verse:
23 And new moon after new moon,
And sabbath after sabbath,
All flesh shall come to worship Me
— said the Lord —
However, the reading begins with an extensive exhortation to pay heed to the spirit, as well as the letter of the Law:
1 Thus said the Lord:
The heaven is My throne
And the earth is My footstool:
Where could you build a house for Me,
What place could serve as My abode?
2 All this was made by My hand,
And thus it all came into being
— declares the Lord —
Yet to such a one I look:
To the poor and brokenhearted,
Who is concerned about My word.
3 As for those who slaughter oxen and slay humans,
Who sacrifice sheep and immolate dogs,
Who present as oblation the blood of swine,
Who offer incense and worship false gods —
Just as they have chosen their ways
And take pleasure in their abominations,
4 So I will choose to mock them,
To bring on them the very thing they dread.
For I called and none responded,
I spoke and none paid heed.
They did what I deem evil
And chose what I do not want.
This is important because the New Moon was a relatively important occasion in Biblical times. It was important to inform all communities about the New Moon so that holidays will be properly celebrated at the same time. Isaiah here points out that it is not sufficient to bring fancy sacrifices -- one must match his ethical behavior to G*dly standards, as well.