The main problem being solved is, of course, that of 'I need to get something done on a bigger screen'. For example, your 5" phone screen is unlikely to let you work productively on an Excel spreadsheet or Word document. In fact, almost everything you do on a phone is compromised to some degree by the size of the display - which is why Continuum (and running the full Windows 10) on the phone is such a potentially good idea.
At the moment, the biggest restriction on Continuum use (aside from having to own one of the half dozen smartphones that support it) is that you need a monitor that either has a free HDMI port or is fully Miracast-ready. As many readers have recounted, many's the time they arrive at a hotel or meeting room and the monitor tech just isn't there.
Happily, things are about to get much easier. For starters, the upcoming (Redstone) Windows 10 "Anniversary Edition" includes a 'Connect' feature which lets any Windows 10 desktop or laptop host a Continuum session. So, in principle, you can 'borrow' the screen of an existing computer to get something done with Continuum on your phone. Why not use the other computer directly? Well, that's an option, but with Continuum you're not disturbing the PC owner's session or authenticated sites - everything's running on your phone, with your ID and your applications and content.
What if that PC is locked, perhaps because the owner is on a lunch break or in a meeting? Well, you'd be stuffed. Except that in the published roadmap (you need to click on the 'In Development' section to expand it), we read:
Touch screen support
Touch input will be supported when connecting a Continuum-compatible phone to a touch monitor.
Laptop-like accessory support
Get support for a new accessory form factor that looks like a lightweight and low-cost laptop, but without any CPU or OS. When paired with a Continuum-compatible phone, it will allow you to use your phone like a laptop.
Projecting on PCs
Instead of only projecting a Continuum for phone experience on unconnected monitors, a Continuum-compatible phone can connect to monitors/screens that are connected to Windows 10 PCs. Connection above the lock screen, as well as behind the lock screen, are supported.
All three are exciting, but let's take them in reverse order. You'll be able (probably in Redstone 2, at the start of 2017) to hook up to other people's desktop monitors and laptop screens, even if they're locked. So you can effect a take over of that display real estate - which brings to mind questions about malicious use, what if the owner didn't want you taking over their screen with your Continuum session? I'm sure Microsoft will crack this before the feature appears, though.
The other two planned features are interesting too, of course. The laptop-like Continuum use case has been seen already, in effect, with the HP Elite X3, so this isn't that new a concept, but it's still very cool.
Finally, having touch interactions on a touch-enabled display interpreted by the OS on your phone is expected behaviour but cool all the same, when it works. It's not really that different after all, to having a Bluetooth mouse connected, but the support for touch over other wireless protocols will be very welcome too.