Show children (sub-navigation) of active menu item only

Ok based on what I think you want from your description, you can accomplish this via CSS. You don’t need to create a custom walker for this unless you have other reasons to do so.

If you look carefully, when a menu item is active it will apply the following CSS to the “li” element

current-menu-item
current-menu-ancestor

So with that information, you can write your CSS rule to show/hide the sub-menus for that “li” item.

For example:

li > ul {
     display:none;
}

li.current-menu-item > ul, li.current-menu-ancestor > ul {
     display:block;
}