You've given an interesting example, among others. Generally this technique is used to accomplish a couple of goals.
First, it keeps the reader interested. When writing a story, it can be unexpected to see the transition between the first-person and third-person view. This can generate a sense of excitement and engage the reader in the text. But this is only minor.
More importantly, it has to do with Isaiah's potential perspective on his command from God and how others should hear God's message. In your first verse (Ch. 6), Isaiah is in personal reflection and searching for something beyond himself. It is natural to use the first-person to describe one's own thoughts.
However, in the next verse (Ch. 7), Isaiah is presenting a command from God to himself. The transition to third-person allows Isaiah to be disconnected from the reader. This allows the reader to interpret a more generic message from the verse. Rather than associating God's command with Isaiah, the reader will simply view the verse as God's command for someone. Surely it is clear that the command is intended for Isaiah, but third-person is far more impersonal. It creates an example of God's calling for people in general, not specifically for Isaiah. Isaiah is essentially saying that any person's name could replace his own to fulfill God's command. If it were in first-person, it would feel personal to Isaiah and imply that God has only called upon Isaiah. Though literally true, Isaiah is trying to make a much more powerful point.