You aren't using any internet bandwidth while watching on your local network through slingbox.com. The stream isn't going out onto the internet...It's just going through your router and then being distributed to your P.C or MAC or whatever hardware you are using to watch the stream on your local network. The slingbox knows if you are either watching the stream remotely or watching from within your home which won't affect your ISP bandwidth whatsoever.
As for the software you mentioned, the only other way in which to watch your slingbox is using the slingplayer desktop software which is a stand-a-lone package allowing you to watch the slingbox without going through the browser. It's no longer supported, but it still works for watching from a local and remote connection and also doesn't use any bandwidth if watching from your local network. If you are using the browser to watch your slingbox stream and it works fine, you don't need this software.