Good question!

For 1 nearest neighbor (1-NN or NN) you can follow the procedure below. For K>1, visualizing the decision boundary by hand becomes quite hard, and is usually done computationally with a grid as in the link you mention. See also here: https://stackoverflow.com/questions/45075638/graph-k-nn-decision-boundaries-in-matplotlib

For (1-)NN:

1. Construct (dashed) lines between the closest pairs of points in different classes.
2. Draw a simple (nondashed) decision boundary for each closest pair (only looking at the points in the pair).
   Step 1 + Step 2 illustrated:
   
3. End the simple decision boundaries where they intersect.
   

Source: http://web.mit.edu/6.034/wwwbob/recitation6-notesfall11.pdf

As a note, the decision boundary for K=1 is equivalent to the Voronoi diagram, which you can find many implementations to draw online. K>1 is more involved and the easiest and non-formal way would be:
1) dividing the region into a grid
2) checking the decision for each point
3) finding the neighbor of points where the decision changes
Best,
Semih