Apple - Quotes problem in Mavericks (or TextEdit)
This is not just TextEdit, it's Mavericks.
Under System Preferences > Keyboard, you can change the quotation marks.
Go to the 'Text' sub-menu. On the right-hand side of the box, uncheck the "Use smart quotes and dashes" checkbox.
The following procedure worked for me:
In TextEdit select all your text by pressing ⌘ Cmd-A and then go to Edit > Substitutions and uncheck Smart Quotes. Make sure Smart Quotes is not active and it should be solved.
If you already unchecked System Preferences > Keyboard > Text > Use smart quotes and dashes.
There's also a checkmark on TextEdit > Preferences > "Smart quotes" (I don't know the translation but it something like that).
Be sure to uncheck that one too.