In the general case, when a figure is translated the x-coordinate of the preimage (ABCD) changes by the value of the horizontal translation a and the y-coordinate of the preimage changes by the vertical translation b.
In this particular case, we are doing a translation of 4 units up, this means that we are translating following the y axis. Thus, to find the new vertices of A'B'C'D', all what you have to do is to add 4 to all y-coordinate of ABCD vertices :
A'(-5,-1+4) B'(-3,0+4) C'(2,-2+4) D'(0,-6+4) which provides finally
A'(-5,3) B'(-3,4) C'(2,2) and D'(0,-2)
Imagine now that they asked you to provide the vertices after a translation of 3 units down and 2 units right.
You will have to add -3 on the y-coordinate and add 2 on the x-coordinate.
You will have to find A'(-3,-4) B'(-1,-3) C'(4,-5) and D'(2,-9)