It's a little more complicated than this and I'm sure others that with experience in PCs and graphics field can chime in, but in a nutshell, I/O devices (a monitor is an output device) in a system have their own range of address space mapped to the device.
Think of the address space like a city: You have special areas in the city that's reserved to perform special tasks, such as city hall. When you need to get a permit, you go to city hall. With the processor, when it needs to update the monitor, it writes data to that address space.
The address space usually points to an external graphics controller, which provides the physical interface to the monitor, provides memory to store data from the main processor, and does housekeeping to keep everyone happy.
When the main processor is ready to "draw" on the monitor, it writes data to that special address space. After the burst of data is sent, the graphics controller takes care of the rest. What happens between the graphics controller and the monitor is a separate subject.
Well, what happens if the processor needs to constantly update the monitor so often that it's tied up doing it and cannot perform other tasks?
That's where a DMA controller comes in. In this case, instead of writing to the address space that's allocated for the graphics controller, the processor writes to an address space that's dedicated to its own memory. After sending out the data to memory, it tells the DMA controller to copy that data to the graphics controller. The processor is then free to do other tasks.
Writing to memory and having the DMA controller copy that to the graphics controller has an extra step, but that extra step is a small price to pay for not having to wait to write data to the graphics controller since that is taken care of by the DMA controller!