The "screen" in your first diagram appears to be an LCD panel with a built-in display controller (a video card, in effect).
It's not just a monitor that takes video signals.
Something along the lines of this, though unlikely to be this exact one:
https://www.buydisplay.com/download/manual/ER-TFTM050-5_Datasheet.pdf
Generally in the circuit sections, all the labelled connection tags just connect to the other tags with the same labels.
You need to realise that the whole thing is a simulation and not a good representation of a practical design.
Edit - I've just had a look at the "nand2tetris" stuff, it's a series of video lectures relating to building a simple computer using VHDL to run in a simulator or programmable logic array.
All the devices and gates are therefore representations of VHDL logic modules. The "display" is a section of RAM bit-mapped to visual pixels.
An overview of the "computer" here, one of the course videos:
https://www.coursera.org/lecture/build-a-computer/unit-5-5-project-5-overview-gjhcz