Student or Learner
The asphalt road gave way about a half mile out to macadam.
Does it mean that the asphalt road stretched for another half a mile until the macadam road started?
Yes, you have the right idea.
This is out of context, so it is just my impression:
To me, "gave way" here suggests that nobody had bothered to extend the asphalt any further, or that they had not bothered to maintain it, because it was a remote rural area. It is as if the asphalt has "given up" and let the more primitive road take over.