Click on a path segment with the Horizontal Straightener tool and it becomes parallel to the top and bottom of the artboard. The neighboring anchor points are shifted to the left and right of the point clicked, moving only vertically. If either of the neighboring anchor points is a smooth point, it's converted to a corner anchor point. Where you click on the path segment determines where the path segment will be redrawn.
After clicking on a path segment, you can drag it vertically to reposition it.
TIP: Hold down the Shift key and the Horizontal Straightener tool becomes the Vertical Straightener tool.
When the Option key (Mac) or Alt key (Windows) is pressed, the Horizontal Straightener tool functions like the Add Anchor Point tool (when over a path segment) or the Delete Anchor Point tool (when over an existing anchor point).
Example: