Why Iterative-Deepening-DFS requires O(b*d) memory?
You need $\Theta(d \lg b)$ memory in order to store where in the tree you are: a position at depth $d$ in the tree is given by $d$ integers in the range $1\ldots b$ indicating for each branch in the tree which particular child you're looking at.

You could think of it as a set of directions of how to get from the root node to the current child. For example: at the root, take the 2nd child, then take the 5th child of that node, and so on.

When you're doing DFS, you need this information: once you've considered all children of a node at depth $d$, you need to go back to the previous node at depth $d-1$ and consider its next child, but you can only get to its next child by knowing what child you're currently looking at.

It's not clear why Wikipedia says $O(bd)$.  Perhaps this is a mistake in the Wikipedia page.