Microsoft Continuum was first introduced last September, as a solution for tablets to seamlessly transition between touch and desktop mode. During today's keynote presentation, Microsoft presented how Continuum has evolved for smartphones with Windows 10.
On a phone running Windows 10, Continuum will offer a (near) full-blown PC if you connect it to an external screen and controls, such as mouse and keyboard. This will allow your smartphone to launch the 'big-screen' version of the UI of the Continuum-optimized app that you're running on your phone, boosting the levels of productivity or entertainment you're getting from it.
During his demo, Joe Belfiore displayed a smartphone's home screen to an external monitor, and then went on to launch Office. Interestingly enough, Office appeared in a form that's very reminiscent of its desktop version on the external monitor, kind of making the Windows phone it was running from seem like a neat, little PC of sorts. Obviously, such kind of desktop-grade UI would be way more comfortable to work on, compared to the limited, touch-optimized experience on the phone's little screen.
In order for Continuum to work as intended, developers need to optimize their apps for this functionality. It was also said that current hardware is not compatible with Continuum for phones (as the feature requires your phone to power not one, but two screens).
source - Microsoft