How are Operating systems involved in execution of programs?
At one level you have hardware: A computer with a CPU, RAM, hard drive, graphics card, monitor, keyboard and so on. 

Then on the lowest level of the operating system you have code that can talk to these devices. That code allows the operating system to read or write data from the hard drive, determine the location of the mouse, and so on. 

At a higher level of the operating system, the OS has code to assign address space to processes, start processes and kill them, allow these processes indirect access to the hardware. 

Above that, it is just code. You have a mouse reporting it's location, you have graphics hardware that can display a cursor, so someone wrote code that keeps track of the mouse location and displays the cursor at the point corresponding to the mouse location. And then someone wrote code to display icons. And more and more and more code on top of that. And that's all, really.