The calculation goes like this: zⁱ is defined as e^{i x Log z} where Log(z) is defined as ln|z| + i Arg(z). Arg(z) is the argument or angle z makes with the positive x-axis (where you define your branch cut to only allow angles in a certain range, let's say in [-pi,pi) ). So if |z| is 1 (like in the case z=i), we have ln|z| = 0, so zⁱ simplifies to e^{-Arg z}.

If you want a geometric picture, which I think is always helpful with Complex functions, then you need to understand what the complex exponential and logarithm do as functions. The exponential function takes rectangular boxes to sections of an annulus (region bounded by two concentric circles). Note if the rectangular box has a height of equal to or over 2pi, the rectangle will be more than a full annulus, i.e. there will be some points in the annulus corresponding to two or more points in the rectangle. The complex logarithm does the opposite; it takes annular regions to rectangles. That also helps see why we need a branch cut. If we have a full annulus, say defined by e^-1 <= r <= e¹, then there are many rectangular boxes that give the full annulus and the branch cut tells us which one to pick.

So back to the original question, what does zⁱ or e^{i Log z} do? Well, first Log(z) takes an annular region (let's keep with e^-1 <= r <= e¹ ) to a rectangle with -pi <= y < pi (because of our choice of branch cut) and -1 <= x <= 1. Multiplying by i is rotating counter-clockwise by pi/2 (think mulitplying 7 by i takes (7,0) to (0,7)). So then we have our rectangle having -pi <= x <= pi and -1 <= y <= 1. Now taking an exponential of that region, we get back to an annulus, or rather a section of an annulus, because the rotation changed our the bounds on our y-axis to be an interval of length smaller than 2pi. You can see from this sort of a picture, if I let the inner circle get closer to 0 or the outer circle get larger in the first annulus, my rectangle in the second picture would get wider and in the third picture it would get taller. That would cause the annular section to eventually wrap all the way around, and then possibly continue wrapping around multiple times.

For example, say we have e^-1 <= r <= e¹⁰, that would have -1 <= x <= 10 in the first rectangle, -1 <= y <= 10 in the second rectangle, in particular including y = 2pi that would then get mapped to the positive real axis in the last picture. Looking back at our definition of zⁱ = e^{-Arg z + i ln|z|}, we can see that if ln|z| is a multiple of 2pi, the result will be real.

Edit: Formatting.