I think many of us where thinking about whats wrong with permissions,even users with Designer permissions will get access denied when creating a site in SharePoint.I encountered the same situation and i was really wondering what is it, then after doing some Google i realized when creating a site (if you have custom CSS and templates) users need read permissions to mater pages and page layouts(Which is located at top level site settings under Site Administration) to create sites.
I hope this will help someone.