Hiya and welcome here.
Congratulations: You found a calculator that can do it very easily and fast without much external assistance.
(although beware of some OS bugs).
Basically, the G0 to G9 graphical objects let you do double-buffering like you see fit and with 10 objects total, you could even do tentuple-buffering
As for screen update speed the reason why it's done so fast in the video is because the calculator's built-in language, HP PPL, is that fast. It can surprise people who are used to HP 50g or even TI-BASIC speed (which can still do very decent stuff), but yeah, you can basically make action games on this platform.
How it's done is another story, though, because it would take me a while to explain and I am a bit busy for that
. There are probably some tutorials and documentation online on how to use DIMGROB_P and BLIT_P commands, as well as that one tool to convert image files to HP Prime PNG format. But basically, DIMGROB_P is the command to setup a graphical object. That object can be any size, so you could easily store several dozens of screens into one (although I usually recommend working with small GROB's such as 336x256 since the calculator can get a bit unstable otherwise), while you could easily just use a tiny GROB to store a few 16x16 icons/sprites.
The BLIT_P command lets you copy a part of a GROB variable (G0 being the screen itself) or part of PNG image data into another GROB of your choice. You can even scale the graphical chunks up or down, although that slows things down.
Here is an example program demonstrating how you can use BLIT_P and DIMGROB_P commands: http://codewalr.us/index.php?topic=124.0
. If you ever figure out how to just shift the screen and only draw the missing rows of tiles to speed things up, then you could write a Mario game. Or you could add some frame skipping to the mix and do Sonic. It's best to start smaller, though
Also, as an advice: If your calculator ever freezes or acts erratically, the ON+SYMB key combo is your friend. It will reboot the calc but you should not lose your programs (backup often, though, like with any other platform!)