Ahaz is reminded that he represents the whole house of David. His attitude is consistent with that of all of Judah. Ahaz and Judah wearied men, surrounding nations were fed up with them, perhaps like many EU nations are fed up with Britain!? But worse than that, they wearied God as well. They needed to turn to God, and God was urging them to do so, and prepared to go to any lengths to get them to do so. That was why He urged Ahaz to seek a sign.
So the Lord would give them a sign, whether they wanted one or not! There is great debate about whether the word translated here means virgin or not, and the debate occurs because of the virgin birth of Jesus. The Hebrew word translated virgin does normally refer to an unmarried woman who would be a virgin. However, in one sense the debate is utterly pointless and demonstrates a misunderstanding of prophecy. Many prophecies, especially messianic ones, have an immediate, or near-future, fulfilment, and a much later the fulfilment. The immediate fulfilment foreshadows the later one. So a son would be born. However, the use of virgin in relation to the immediate virgin just means the woman was not married yet. So there was no “virgin birth” in Isaiah’s day. When Jesus came there was a virgin birth (or more properly, perhaps, a virgin conception). Immanuel means “God with us”.
Now let’s look at what is going on here. Ahaz, representing Judah, is refusing to live by faith, refusing to have much if anything to do with God. He is denying God’s existence or relevance. Even so, God will send a sign that He is indeed with them, whether they like it or not. The world as a whole denies God’s existence or relevance, especially in the West. We can be sure that at some point God will intervene, and demonstrate His existence and relevance!