Enable Edit and Continue option is useful when your web application is running on local IIS Express server. You can find this option from Web Application Properties -> Web Tab-> Debuggers -> Enable Edit and Continue.
Fig :Default 'Enable Edit and Continue' option is Checked in VS2013
Visual Studio 2010/2012 - Enable Edit and Continue option is un-checked by default when a new web application project created. But where as with visual studio 2013, the option is checked by default.
From the image, by default "Enable Edit and Continue" option is checked and under servers "IIS Express" is selected. If you change the server to 'Local IIS/External Host', the option becomes disabled mode.
Enable Edit and Continue ON - When you stop debugging, IIS Express server will be closed and you can find the same in windows task bar, IIS Express icon will be gone. With this if you try to run "http://localhost:40740"(URL refereed from image) you will get "could not connect to localhost:40740".
Enable Edit and Continue OFF - When you stop debugging, IIS Express server is still running and windows task bar has IIS Express icon. If you run "http://localhost:40740" , you can see your web application in browser.
Note: If you want to run your web application even after debugging is completed, un-check the Enable Edit and Continue option.