How to browse the server to upload files from there in Drupal 7
****UPDATE****

Best option is to use https://drupal.org/project/imce_filefield
This is specially made to upload files to sites/default/files and add them.

Just configure imce and the field.

Uninstall file field sources, because it's not compatible to have them at the same time



Old anwser

You could use the FileField Sources module. To browse the server it looks like you need to use this with IMCE. From the FileField Sources project page:

The FileField Sources module expands on this ability by allowing you to 
select new or existing files through additional means, including:

    Re-use files by an auto-complete textfield
    Attach server-side files uploaded via FTP
    Transfer files from a remote server (cURL required)
    Select existing files through the IMCE file browser


This question / answer was on another question:
Image Field Browser Option