Might be a dumb question but did you download the patches? I think one of the patches enabled the mods option.
I'm on the XBox ONE console and this is what I get.
Disc installed to HD, initial update (around 500mb) installed.
Connected to XBox LIVE. Turn on XBox and select game.
First screen is Bethesda logo on the bottom left, Skyrim symbol in the centre of the screen and it says "Press Any Button".
Press a button and then you get:
Your profile pic and name in top left, Bethesda Logo in bottom left, Skyrim symbol in the centre of the screen and the following menu on the bottom right:
Continue
New
Load
Add Ons
MODs
Credits
Help
You select MODs and have to log into Bethesda.net and then can peruse and add.
If you do not have MODs in your menu here then...
Do you have XBox LIVE or PSN Plus?
If you do not they may be required to enjoy the MODs section of the game.
Do you have LIVE or PLUS?