This question on the English Language Stack Exchange site talks about the Multiple Quotation Rule.
When a speech goes over more than one paragraph then the opening quotation marks are repeated at the beginning of each paragraph, with closing marks placed only at the end of the speech.
This is not a rule universally followed but is followed here in the NIV.
The NIV are also following another convention of indenting longer sayings.
3Then the Lord said to Isaiah, “Go out, you and your son Shear-Jashub, to meet Ahaz at the end of the aqueduct of the Upper Pool, on the road to the Launderer’s Field.
4Say to him, ‘Be careful, keep calm and don’t be afraid. Do not lose heart because of these two smoldering stubs of firewood—because of the fierce anger of Rezin and Aram and of the son of Remaliah.
5Aram, Ephraim and Remaliah’s son have plotted your ruin, saying,
6“Let us invade Judah; let us tear it apart and divide it among ourselves, and make the son of Tabeel king over it.”
7Yet this is what the Sovereign Lord says:
“ ‘It will not take place,
it will not happen,
8for the head of Aram is Damascus,
and the head of Damascus is only Rezin.
Within sixty-five years
Ephraim will be too shattered to be a people.
9The head of Ephraim is Samaria,
and the head of Samaria is only Remaliah’s son.
If you do not stand firm in your faith,
you will not stand at all.’ ”
The opening double quote in verse 3 is repeated at the start of the indentation in verse 7, and closed at the end of verse 9.
The single quote in verse 4 is also repeated at the start of the indentation in verse 7, and closed at the end of verse 9.
The double quote at the start of verse 6 is closed at the end of verse 6.
The indentation in the middle of verse 7 counts as an opening quote and this closes with the end of the indentation at the end of verse 9.
Verse 3 begins The Lord said to Isaiah - the rest of verse 3 and all of verses 4,5,6,7,8 and 9 are what the Lord said to Isaiah.
The Lord's message to Isaiah was to go and find Ahaz and give him a message. From after "say to him" in verse 4 until the end of verse 9 is what the Lord wants Isaiah to say to Ahaz.
The message Isaiah is asked to give to Ahaz is to tell him what Aram and co. have been saying (verse 6) but not to worry because the Lord says something else (the indented bit from middle of verse 7 to end of verse 9.
The Lord told Isaiah to go and tell Ahaz something. Isaiah was to tell Ahax what Aram said and also what the Lord had to say on the matter. So the indented portion is what the Lord is telling Isaiah to tell Ahaz what He, the Lord has to say about the matter. (The doctrine of verbal inspiration would even further complicate things.)
The Lord said to Isaiah,
Go to Ahaz and say to him
Be careful, Aram and his pals have been saying
Let us invade Judah
Yet this is what the Sovereign Lord says:
It's not going to happen.