There are many videos on YouTube explaining just how to use it, some making it far more complicated than necessary, but I found the easiest way to remove an object is to use the lasso tool, surround the object, as close as possible, that you want removed, hit the delete button, enter in the "use" box content aware and then click okay, and you're done. You can do the same thing with the rectangular marquee tool. You can also use the spot healing tool, which will remove the object without having to press delete. Remember that the closer to the object the better. The closer the object is to other objects, the trickier it gets.
Here is an example: I loved the photo of this poodle sitting outside a bakery in Old Quebec City, but I found the silver electrical outlets very distracting, so I decided to try the Content Aware tool.
This is the result:
Pretty darn amazing, and it was very easy.
For more detailed instructions and different ways of using the feature, check out the following articles and videos:
There are a ton more on YouTube.