You need that box to convert the electricity from AC to DC or vice versa. The converter plug usually has a transformer box with an adapter in one piece. If not then you would have to buy an adapter to plug into the wall then plug your converter into the adapter then plug-in your US bought device. The reason why you would buy this is for your devices that don't have a box (i.e. your straightener). If you simply use an adapter with your American bought straightener then it'll blow up or do something else unpleasant.
The Voltage Converter shown in the link you posted should not be with electronic products, such as those mentioned by the OP. That type of Converter uses solid-state technology to reduce the voltage, rather than a Transformer and the output is not especially "clean". When connected to a solid-state product such as a Camera Charger, unfortunate consequences can occur! Most manufacturers of electronic products such as cameras specifically recommend against using their products with that type of Converter.
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