Could somebody explain to me why the height of a weight balanced binary tree in $O(\log n)$ in the worst case?
What have you tried? Where did you get stuck?
In particular, you're going to want to look at a proof by induction. Does a tree with 1 node have logarithmic height? Then look at a tree with $n$ nodes. When you add another node to a balanced binary tree, which you know to have logarithmic height, what is the worst thing that can happen to the height?
Think of a Weight Balanced Tree (wbt) as a certain instance of a quick sort run. The invariant of the wbt indicates that the 2 halves of the sub-arrays to be sorted recursively are Theta of each other. This means that each tree of execution in quick sort will never be deeper than Theta(log n).