When you have opened the extsdcard folder in airdroid, you can click on the blue button in the upper right called 'upload'. This will open a new window where you can drag & drop files to add them to the sd card. To delete files, you can just highlight them in airdroid and hit the delete button on your keyboard. Airdroid does not recognize how much space is on the external sd card for me, so if it is the same for you you will have to track your free space on the phone itself by using a file manager or going to settings -> storage and scrolling down to the sd card area.
Ignore the warning about system files, there are no important files on your extsdcard. Just don't mess around with the other files in the system folder unless you know what you are doing.