Ban closed his eyes and filtered the noise so that he could only hear the area around the tree where he hung the roasted meat. As of the moment, the meat was still there, unattended.
Ban already guessed that, that woman should be the guard that Kuys assigned to look out for Ban. He knew that she had already caught up, and might have only been hiding to avoid alerting Yakob and the others.
He considered that as she was looking out for him, she must not have the time to hunt for her own food. He was considerate enough to give her something to eat.
Ban also did not make this move out of the blue and just because he pitied her. He was actually using this as some sort of test to determine her real identity. If she was truly a guard assigned by Kuys, he knew she would not hesitate to eat that mate he gratefully offered.