Once your Archive contains at least one drawer, you can start adding boxes to it that you want to archive. Note that you can only add boxed up items to the Archive.
If you already have a box, you can simply Ctrl+drag it to the Archive, and it will ask you which drawer you want to add it to. If you own the Wardrobe, it might also ask you if you want to associate the box with a Wardrobe item.
If you need to box up items first, the Archive can rez a basked for you into which you can drop the items. Touch the Archive and select "Rez Basket" from the menu and type in a name when it prompts you. Fill the basket with your items by Ctrl+dragging them onto it.
Alternatively, if you're using the Firestorm viewer, you can zdrop items into the basket. The basket chats the command in local chat, so you can just copy and paste it, substituting the correct folder name. (It is useful if you use the name of the folder you want to box up for the basket, then the zdrop command is already correct.) Note that, for zdrop to work, the folder must be a top-level folder, i.e., contained directly below the "Inventory" root folder. You might have to move it there first before using zdrop.
Once everything is in your basket (you'll notice that you'll get a visual indication that it contains stuff), take the basket into your inventory, and from there, ctrl+drag it onto the Archive.