As an older brother, Klaus has certain duties to uphold. His life experience makes him responsible for teaching his younger sister the ways of the world. He teases her in order to make her strong. He gives anyone suspicious “the look.” And, if she is in trouble, it is his job to rescue her, no questions asked.
In this case, Estraeah is not in danger. However, she has run off again, and it is all his fault. His guilt leads him to follow.
Doubtlessly, she has ventured into the forest, as she always does whenever her flighty nature takes hold. Klaus knows she will be upset, but it always worries him when she goes off without warning. And so here he is, following the well-manicured path Estraeah has cut out over the years.