If the equation were (z+4)^3 = -8, you would just take the cube roots of both sides of the equation and then subtract both sides by 4 to get z by itself. With the equation in the problem, you will still do those two steps. The only difference is that finding the cube root of -8i requires a bit more work than finding that of -8.
It may be helpful to think of rotations in the complex plane. Imagine having two points in the complex plane and drawing a line from the origin to each of the points. Multiplying the two complex numbers multiplies the lengths of their lines together and adds together the angles each line has from the x axis. So if, for example, you took a complex number that was located at a distance of 3 from origin and at 40deg angle from the x axis, multiplying that number by itself would result in a number with distance of 9 from the origin and at an 80 degree angle from the x axis.
On the complex plane, the number -8i is at at a 270deg rotation from the x axis. We need a number or numbers on the complex plane that when multiplied by itself 3 times gives you -8i. The distance of that number from the origin is going to be 2, since 2*2*2 =8. We just need the angle (or angles) that when added to itself 3 times gives you 270deg. The obvious solution is 90, since 90+90+90=270. So one answer to the cube root of -8i is 2i (and you can easily check that 2i*2i*2i=-8i).
However, that is not the only solution, because if there are any angles between 0 and 360 that when added to themselves 3 times gives 270+360n for a whole number n, then that would also work. For example, 270deg is the same as 630deg, since 270+360= 630 (or in other words 630deg = 270+1 full rotation). So 630/3 =310deg also works. There's also one more angle that you can find this way, but using this angle as an example: to find the actual complex number, we can go from polar coordinates (using the angle and distance) to rectangular coordinates (x+yi), using the formulas x=rcosθ, y=rsinθ, where r here is just equal to 2, the distance of the point from the origin. Using these, you should be able to get all three cube roots of -8i. Then, just remember to subtract 4 from the result, and those will be your solutions to the equation.
I hope this helps clear things up. If you have any other questions please feel free to ask. Thank you for your time and have a wonderful day.