First consider y=tan(2x-4) compared to y=tan(x).
When the argument is zero, y is also zero, so y=tan(x) passes through the origin (0,0). This is one of the points of intersection of the graph and the x-axis. This point of is shifted to (2,0) because 2x-4=0 when x=2. The graph also intersects the x-axis when x=π for y=tan(x) and when 2x-4=π, that is, x=(π+4)/2. Because the tangent function is cyclic, there are many points related to these intersection points. While the intersection points are nπ (n is an integer) for y=tan(x), those for y=tan(2x-4) are when 2x-4=nπ, that is, when x=(nπ+4)/2. The separation of each intersection for y=tan(x) is π, while that for y=tan(2x-4) it's π/2.
Now consider y=tan(2x-4)-3. There is a vertical displacement of 3 units, so these intersection points are shifted downwards as well as sideways.