That is because once connected to the AEBS it becomes a network drive. You then have to mount the network drive in order to use it.
You should not change the permissions attached to the drive after you have partitioned and formatted it. Be sure you have prepped the drive for this use:
Drive Preparation
1. Open Disk Utility in your Utilities folder.
2. After DU loads select your hard drive (this is the entry with the mfgr.'s ID and size) from the left side list. Click on the Partition tab in the DU main window.
3. Under the Volume Scheme heading set the number of partitions from the drop down menu to one. Click on the Options button, set the partition scheme to GUID then click on the OK button. Set the format type to Mac OS Extended (Journaled.) Click on the Partition button and wait until the process has completed.
4. Select the volume you just created (this is the sub-entry under the drive entry) from the left side list. Click on the Erase tab in the DU main window.
5. Set the format type to Mac OS Extended (Journaled.) Click on the Security button, check the button for Zero Data and click on OK to return to the Erase window.
6. Click on the Erase button. The format process can take up to several hours depending upon the drive size.
Eject the drive upon completion and connect to the USB port on your AEBS. Open Airport Utility to configure it under Drives and turn on Sharing and File Sharing. You can then mount the drive for use by selecting Connect to Server under the Finder's Go menu.