The home page slideshow uses nodes of type Slideshow Image.
To add an image to the slideshow, create a new node of type Slideshow Image:
Content -> Add content -> Slideshow Image
Images should be horizontal (in "landscape" mode), no smaller than 600 pixels wide and 400 pixels high, and 72 ppi (pixels per inch).
To remove an image from the slideshow, find the node for that image, and unpublish it.