You may want to read the WD My Cloud User Manual (http://www.wdc.com/wdproducts/library/?id=439&type=25) that explains several methods of accessing content stored on the My Cloud.
One can access data file content on a My Cloud (when remote access is enabled) using the MyCloud.com web portal. Use of the web portal will require some configuration on the WD My Cloud using the My Cloud Dashboard administration interface. One can use the Dashboard to configure individual users and restrict which folders (called Shares) they have access too via passwords. The User Manual explains how to create both users and shares on the My Cloud.
For media files there is the Twonky media server (include with the My Cloud) administration interface that will also allow one to browse and play media files stored on the WD My Cloud.
Other than those options, FTP, SSH, the WD My Cloud Desktop software application and the WD mobile apps there isn’t much of a way to access the WD My Cloud via a web browser as opening the WD My Cloud with a web browser generally takes one to the Dashboard administration interface. It may be possible (by violating the warrantee on the device) to use SSH to upload a separate web server module or modify the internal My Cloud web server to allow for some type of general browser access that doesn’t load the Dashboard UI.