You add a web part to the page in SharePoint 2013, but are unable to edit it because you only see an arrow that shows/hides the web part.
The cause for this issue is you are using Internet Explorer 11 with SharePoint 2013.
You have a few options:
~ Use Chrome with SharePoint 2013.
~ Remove the update for Internet Explorer 11 and use Internet Explorer 10 or lower version.
~ Change the SharePoint 2013 page you are viewing in compatibility mode to Internet Explorer 10 or Internet Explorer 9 on your computer.