This is not new to Office 2013, but apparently many users skipped Office 2010 and are only now discovering that some older files won't open in the Office applications. This is because of a security feature called File Block.
The File Block Settings will force the selected file extensions to open in Protected View or will block them completely. If you run into problems opening email attachments, you need to change the settings in the application that will open the attachment, not in Outlook.
To access the File Block Settings, open the application (Word or Excel etc) and open it's File, Options, Trust Center dialog. Select File Block Settings to view the settings.
Any file types selected in the Open column (1) are controlled by the selection at the bottom of the dialog (2).
Any file types that are selected in the Save column cannot be saved. You'll need to Save as in a different format.
While you are in the Trust Center, check your Protected View settings. While Protected View won't prevent you from opening files, it will require an extra steps or two if you want to edit the file. You'll need to change the protected view settings for email attachments in the program that is used to open the attachment, not in Outlook.