The Awesomeness rating is an obscure statistic in Dead Rising 3.
Its only use is for the survivor search mechanic. When the player tells a survivor to search the area, the survivor will go to pick up the weapon with the highest awesomeness rating. If the weapon they are holding is higher or equal to any weapon in the area's awesomeness rating, they will simply say that they couldn't find anything of value. This does not apply to food items.
There is no way to view a weapons awesomeness rating except for directly finding it in the weapons coding.