[UPDATE]
The original question, and the answer below applied specifically to the IE11 preview releases.
The final release version of IE11 does in fact provide the ability to switch browser modes from the Emulation tab in the dev tools:
Having said that, the advice I’ve given here (and elsewhere) to avoid using compatibility modes for testing is still valid: If you want to test your site for compatibility with older IE versions, you should always do your testing in a real copy of those IE version.
However, this does mean that the registry hack described in @EugeneXa’s answer to bring back the old dev tools is no longer necessary, since the new dev tools do now have the feature he was missing.
[ORIGINAL ANSWER]
The IE devs have deliberately deprecated the ability to switch browser mode.
There are not many reasons why people would be switching modes in the dev tools, but one of the main reasons is because they want to test their site in old IE versions. Unfortunately, the various compatibility modes that IE supplies have never really been fully compatible with old versions of IE, and testing using compat mode is simply not a good enough substitute for testing in real copies of IE8, IE9, etc.
The IE devs have recognised this and are deliberately making it harder for devs to make this mistake.
The best practice is to use real copies of each IE version to test your site instead.
The various compatiblity modes are still available inside IE11, but can only be accessed if a site explicitly states that it wants to run in compat mode. You would do this by including an X-UA-Compatible header on your page.
And the Document Mode drop-box is still available, but will only ever offer the options of “Edge” (that is, the best mode available to the current IE version, so IE11 mode in IE11) or the mode that the page is running in.
So if you go to a page that is loaded in compat mode, you will have the option to switch between the specific compat mode that the page was loaded in or IE11 “Edge” mode.
And if you go to a page that loads in IE11 mode, then you will only be offered the ‘edge’ mode and nothing else.
This means that it does still allow you to test how a compat mode page reacts to being updated to work in Edge mode, which is about the only really legitimate use-case for the document mode drop-box anyway.
The IE11 Document Mode drop box has an i icon next to it which takes you to the modern.ie website. The point of this is to encourage you to download the VMs that MS are supplying for us to test our sites using real copies of each version of IE. This will give you a much more accurate testing experience, and is strongly enouraged as a much better practice than testing by switching the mode in dev tools.
Hope that explains things a bit for you.