Assuming you know the graph of tan(x), let's analise step by step what changes you make, and their consequences.
First step: from tan(x) to tan(x-\pi). This changes is one of the form f(x)->f(x+k). This kind of changes means a horizontal translation of k units, to the left if k is positive, to the right if k is negative.
So, in you case, we start from the graph of tan(x), and shift it to the right of \pi units.
Second step: from tan(x-\pi) to -tan(x-pi). This is one of the most simple and intuitive changes: if you go from f(x) to -f(x), you simply change the sign of every single value of the function, resulting in a reflection with respect to the x-axis.
Third step: from -tan(x-pi) to -tan(x-pi)-3: this is a change of the form f(x)->f(x)+k. This means that you need to add a certain number to the values of the function, resulting in a vertical shift, which will be upwards if k is positive, and downwards if k is negative. So, in your case, you need to shift the graph of -tan(x-pi) down of three units.
