My guess is that you didn't get the HD towing package for your Explorer.
Here is a
link to the 2013 Ford Towing Guide.
Without the package, you only get 2000 lbs max tow rating. With the package (and using a WD hitch kit), you get 5000. 2000 lbs may be enough for a VERY small utility trailer, but keep in mind, that 2000 lbs capacity includes any passenger and cargo weight you put in the Explorer. 3 passengers could reduce your effective tow rating to 1500 lbs.
Page 12 shows that the HD tow package on the Explorer (look for option 52T on the sticker), would have included a wire harness (to 4/7-Pin connector), the hitch receiver, and extra engine oil cooling. So the fact that you had to add a hitch proves that you don't have the towing package.
My recommendation:
1. Get that kit that etrailer recommended. I use 1 of these on my Chrysler minivan when I was pulling a popup camper. These are very common kits that easily tie into the lighting system with minimal work. Some tie into existing connectors, and others splice into wires. Get whatever etrailer recommends. The instructions will be thorough. Then again, if the plug-in kit will plug into that dangling end, and provide lights, who cares if the rest of the wiring isn't there. Maybe you only need to test that dangling plug to see if there is a signal there when the lights are on. You'll need to find which wire is ground, then check the other 3.
2. You already have the hitch.
This will give you the ability to use that Class I 2000 lbs MAX towing capacity. But like I said, that full 2000 lbs is only available if the Explorer is empty (driver only).
If you think you need/want more than the 2000 lbs, IN MY OPINION, adding that engine oil cooler would bump you up to a Class II 3500 lbs MAX tow rating. Check with the dealer. It's probably a simple in-line cooler that can be added.
To get the full 5000 lbs rating, per the towing guide (page 20) you would then need to use a weight-distributing hitch (WDH) kit. But that's for very large utility trailers or campers.