For a role to have the ability to Export (download) the Report results, Saved Searches results, and Lists results, the role must have the Export Lists permission.
1. Login as Administrator and navigate to Setup > Users/Roles > Manage Roles.
2. Edit the role.
3. Under Permissions tab > Lists subtab > add Export Lists.
Note: Only available Access Level is Create.
4. Click Save.
For more information about exporting, please see Exporting Reports, Searches and Lists, Answer Id: 21884.