When Fallout 4 mods come to Xbox One, there will be a few restrictions.
Mods are coming to the Xbox One version of Fallout 4, but, they won't work exactly like the wild wild west of PC modding. The XB1 is a closed platform owned by Microsoft, afterall, so there will be some restrictions. In a live stream event Bethesda detailed some of those restrictions, the most significant of which being that mods for the Xbox One version of Fallout 4 have to be smaller than 2GB.
Installing the mods themselves is fairly simple: you just have to link your Bethesda account to the Xbox version of your game, then you can browse the mods from the game's main menu. You can also check out available mods online here. Due to first-party considerations, you cannot earn any achievements while you have mods enabled, and any mods that use assets from DLC require players to have purchased said DLC.
The game will also create a separate save when you first start using mods, so if something goes wrong you can revert to your non-modded save.
Mods for the Xbox One version of Fallout 4 will be released on May 31. PlayStation users still don't have a solid release date yet, but Bethesda claims it is still "on track" with it's initial goal of bringing mods to the PlayStation 4 in June.