The WD device would connect to your network/Internet either via wireless or by cable. This then allows it to connect to your Slingbox and receive the stream, which it then puts on the TV.
By cable means it would use an ethernet cable plugged into your router instead of wireless, or if your router is too far away then you can use a technology called 'HomePlug'. I use this and it's brilliant for me but depends on the quality of your electric wiring in your house. Basically you buy two of these HomePlugs and one plugs into the mains next to your router and then a network cable connects it to your router, and the other plugs into the mains next to your WD device and then a network cable connects it to the WD device. This essentially creates the equivalent of you running a long cable from the WD device to your router and i find it better and more reliable than wireless.
However, I would recommend trying with wireless first, and only using HomePlug if you need to, ie if the signal is unreliable.
When you set things up as per the above, you can use any TV, ie it doesn't need to be an Internet ready one. The WD device as described above connects to your network/Internet so the TV just needs to be able to connect to the WD device via HDMI.
An alternative approach would be to use an iPad. If you buy the Slingplayer app for the iPad this will then be able to view your Slingbox on the iPad. You could then buy an Apple TV and then use a technology called Airplay (which both iPads and Apple TVs come with) and this allows you to project what is your iPad screen to the TV connected to the AppleTV. The Apple TV is a small black box that you would then need to connect to your network/Internet either wirelessly or as described above using Homeplug.
Both are viable methods, but I'd recommend scouring these forums for the latest state of play regarding both methods as Sling have a habit of breaking functionality with their compulsory firmware updates and I've not personally tried either for months.