It seems to be about having a PC wherever you are as long as you brought a mouse and keyboard with you and also have access to a TV with proper inputs. I am pretty sure most people who have a smartphone will also have a laptop or tablet in 2015. I think MS just needed an idea to push windows10 mobile and went with this, for whatever reason..
You actually don't need a keyboard and mouse - the phone's touchscreen can be used as keyboard/mouse.
TV with inputs - interestingly most new "smart TV" that you can buy now comes build in with a Miracast reciever, ergo this is because a lot of the TV makers (Samsung, LG, Sony etc) also happens to be smartphone makers and they want to copy Apple's much praised "magic" of Airplay thus the inclusion of Miracast as standard TV equipment. As Miracast is an industry standard so eventually having the right input shouldn't be a problem anymore.
Finally Continuum is one of those "the best camera is the camera you always have with you" things - sure anyone can buy a good mirrorless or DSLR that would blow your phone camera away in terms of features such as controls, lens selection etc, but then people now vastly prefer to shoot photos on their phone because of the convenience of it, not because they are looking for the "best tool for the job". Continuum is going to be that analogy for the smartphone/PC/laptop thing for a lot of the not so technical people.