The magic you see there is a combination of many heavy technologies. HTML DOM objects floating freely, moving by script inside a JIT script runner inside a browser, the task of dragging and sliding requires a relatively powerful computer to make it look smooth.
The smooth UI of moving large boxes around like that is relatively recent with modern processing power. Back in the Win95 and Win98 days a company I was at was trying to build similar custom controls, and the animations would typically peg the CPU as small boxes were moving. We recommended at least a 600MHz Pentium 3 as a minimum spec for smooth display.
There is nothing built in to Windows that looks similar. The tech built in to the API is extremely lightweight, mostly about forwarding events and data packets that your program can respond to, the core of the Windows API stuff was built for processors from the 1980s and 1990s.
For implementation, options include creating your own owner-drawn component with movable boxes inside it, creating small boxes that respond to events and push themselves up and down, or otherwise doing much of it yourself.
Drag and drop could work, but it is normally less visible than you describe. For Windows drag and drop typically only the mouse cursor changes when you are dragging, and the application may do some extra paint calls to perhaps drop a marker where the drop will take place. Whatever drawing you do is also custom work. It would not look like what you see in jQueryUI. If you are using any of the tools like WinForms to build your UI it may be a little easier since it is visually on screen as you edit, but still will require some work.
Another option is to embed some HTML viewer in your program. A very heavyweight solution, but it can be done.
Another option is to buy a controls library, like those sold by DevExpress, which was the way things were often done before the modern web browsers and web apps. Most of these controls cost several hundred dollars. It may be a lot for a hobby developer, but for a business that is cheaper than paying a worker for the weeks it would take to develop, refine, debug, and generalize the solutions.
Whatever you end up going with, good luck on it.
