It is called "connect the dots."
In adult discourse the phrase "connect the dots" can be used as a metaphor to illustrate an ability (or inability) to associate one idea with another, to find the "big picture", or salient feature, in a mass of data. - The dubious Wikipedia
The idea is; as you collect seemingly relevant evidence, you connect the dots to the other evidence where it crosses over. Eventually you are able to see the big picture of what is really happening (or at least where to go to look for more clues).
When a person literally does this on a pushpin board, they are creating a visual reference for the evidence they have gathered.
As to why it is red, it is probably because it would usually have a high contrast relative to the information on the board.