How do I enable the browser spell checker in WYSIWYG editor?
The Check Spelling plugin is only included with CKEditor Standard and Full by default.

For the CKEditor module spell checking is enabled via the profiles.

Go to Configuration > Content authoring > CKEditor and edit each profile you to to it display on. Then under Editor Appearance drag the appropriate buttons to the "Current toolbar"

If you are using the WYSIWYG module it is also enabled through the profiles.

Go to Configuration > Content authoring > Wysiwyg profiles and edit each profile you want it to display on and check the "Check Spelling" option.

With CKEditor 4 you must enable "Spell check as you type" since the buttons have been merged into one with WYSIWYG.