You shift every point 6 units to the right and 6 units down. Check out the attached image below. In that image, I show point A moving to point A' in this fashion. Every other point (B and C) will move along this similar pattern. This is how the triangle ABC moves to A'B'C'
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In terms of an algebraic rule, it would be (x,y) --> (x+6, y-6). We add 6 to the x coordinate of each point. At the same time, we subtract 6 from each y coordinate.
For instance, A is at (-2,3). Add 6 to the x coordinate and we get (4,3). Now subtract 6 from the y coordinate and we end up with A' = (4,-3). Points B and C will be treated in a similar manner.
Note: Triangle ABC is called the pre-image ("pre" meaning "before") while A'B'C' is called the image.