I have downloaded his repo to build my own spin of a 8-bit computer....Well I've downloaded several repos and I am picking out bit I like out of each.
Also I'll design and build the bits that I don't like to be how I like.
Those dots are express sketch "pins". They represent connection points for the component. The pins have properties like number and function and the properties are editable by clicking on the pin.
I see a breadboard assembly as transient - so I'd work from a schematic.
"protoboard" (veroboard) is different, a more stable and reliable prototype, and I would usually plan the layout on graph paper (generated in excel to suit) with a pencil.
or one could use VeeCAD which ties the layout and schematic together. http://veecad.com
I did look at that thread briefly. It does look like you had the led the wrong way round initially. The "bent knee" is positive.
But why oh why don't they add a '+' to the graphic to make it unambiguous? Anyway, that may have contributed to the big mess. However, it is clear from your research that we can't soon expect to see immaculate schematics emerging from Fritzing breadboard layouts.
Also, for analog circuits, there is also LTspice which is actually a simulator but has a quite simple and intuitive schematic builder. This is in its "Print Preview" mode: