Access denied when attaching new media item.

When a Sitecore non-admin user tries to attach a media item, access denied error appears. 

This is due to incorrect Security settings in Sitecore. My Sitecore Version:  8.1 (rev. 160519)

Solution by Kasaku. 

3272 11:36:34 ERROR Application error.
Exception: System.Web.HttpUnhandledException
Message: Exception of type 'System.Web.HttpUnhandledException' was thrown.
Source: System.Web
   at System.Web.UI.Page.HandleError(Exception e)
   at System.Web.UI.Page.ProcessRequestMain(Boolean includeStagesBeforeAsyncPoint, Boolean includeStagesAfterAsyncPoint)
   at System.Web.UI.Page.ProcessRequest(Boolean includeStagesBeforeAsyncPoint, Boolean includeStagesAfterAsyncPoint)
   at System.Web.UI.Page.ProcessRequest()
   at System.Web.UI.Page.ProcessRequest(HttpContext context)
   at System.Web.HttpApplication.CallHandlerExecutionStep.System.Web.HttpApplication.IExecutionStep.Execute()
   at System.Web.HttpApplication.ExecuteStep(IExecutionStep step, Boolean& completedSynchronously)

Nested Exception

Exception: System.Reflection.TargetInvocationException
Message: Exception has been thrown by the target of an invocation.
Source: mscorlib
   at System.RuntimeMethodHandle.InvokeMethod(Object target, Object[] arguments, Signature sig, Boolean constructor)
   at System.Reflection.RuntimeMethodInfo.UnsafeInvokeInternal(Object obj, Object[] parameters, Object[] arguments)
   at System.Reflection.RuntimeMethodInfo.Invoke(Object obj, BindingFlags invokeAttr, Binder binder
   , Object[] parameters, CultureInfo culture)
   at System.Reflection.MethodBase.Invoke(Object obj, Object[] parameters)
   at Sitecore.Web.UI.Sheer.ClientPage.OnLoad(EventArgs e)
   at Sitecore.Shell.Applications.Dialogs.Attach.AttachPage2.OnLoad(EventArgs e)
   at System.Web.UI.Control.LoadRecursive()
   at System.Web.UI.Page.ProcessRequestMain(Boolean includeStagesBeforeAsyncPoint, Boolean includeStagesAfterAsyncPoint)

Nested Exception

Exception: Sitecore.Exceptions.AccessDeniedException
Message: Application access denied.
Source: Sitecore.Kernel
   at Sitecore.Diagnostics.Assert.HasAccess(Boolean accessAllowed, String message)
   at Sitecore.Shell.Applications.Dialogs.Attach.AttachForm.OnLoad(EventArgs e)