Its a garbage UI in this regard and highly confusing, and lacks needed details and specficity.
This is nothing new in Behesda productions. This will be fixed later, by modders once they have the tool set.
I hope, and expect anyways.
One example, a proper sorting method for varous catagories, like guns for example. THe entire naming scheme of the guns and default sorting
method of that catagory is pathetically bad. The obvious sort thing missing you ask? By caliber and type as type defined as pistol or rifle or other.
How that got past testing and quality control is beyond me as they claim to have spent 7 years on this.
FO NV and FO3 had similar issues that were later fixed by the mod community.
To actually answer the question: Mods in your inventory are waiting to be put on a weapon.
As for removing mods, you need to select a different mod for that slot. The game applies it, and the old mod is put into your inventory. If you already have that mod, the option at the bottom changes from 'Build' to 'Attach', and activating that mod will simply attach the indicated mod to your weapon from your inventory.
Actually I found that mods in your inventory couldn't be used on stations, they need to be put in the Workshop inventory to you have access to them on the crafting station very weird.
Wait....I thought it was the exact opposite? As I recall, I had all my mods in the Workshop storage and I couldn't apply any of them to weapons. I recall having to transfer them into my inventory to actually apply them at the bench. Will need to retest this as it SHOULD work either way.
EDIT: Tested this last night and it works both ways for me (PC).
You can only add a mod to the weapon at the weapon bench, but you can deposit or withdraw at any station. I tested this with both a shotgun, combat rifle and 10mm pistol.
Note this is based on a workbench / stations you own as part of a settlement. I've not done much testing with workbenches or stations you do not own (as all my crafting resources are at other workstations).
LoL works for me by putting them in the Workstation not in my inventory *shrugs* and yeah should be both, agreed.
That's the thing, in a settlement you own, the R key, whatever crafting station you're at, opens up Workbench. So it soesn't really matter. I remember trying to mod a weapon on a workbench outside a settlement and it wouldn't allow me to use mods that I had in my inventory. I must have tested it out in my settlements at some point because I ended-up putting all my mods into the workstation. *shrugs*
If you already have the mod, the workshop UI will say "Attach" the mod instead of "Build" it, so I don't think it's confusing at all.
To answer OP's question, your inventory shows UNUSED mods.
Because it is almost a necessity to keep up offense and defense numbers? I rarely find loot with mods beyond my ability to create and even when i do they generally need to be upgraded in some other slot to really stand out. if i wasnt modding gear then my defense would be at about half its current and my weapons would be doing around 1/3 their current damage.
Because you can craft a weapon that matches your play style and requirement much better than just using what you pick up. You can also turn a Legendary weapon with a great bonus into a weapon that is much more useful to your desired combat style.