Think of the logarithm as giving you back the angle between the positive real axis and the number you have plugged in (times a factor $i$). You were not surprised to see
However, this statement just means that the angle between the positive axis, and the negative axis (represented by $-1$) is $180^°$ or (in radians) $\pi$. And when you look at the complex plain, you would see that $i$ is "above" the zero, hence there is a $90^°$ angle between the positive axis and the complex axis (represented by $i$), which expressed by radians is $\pi/2$. Hence
$$\log(i)=\frac\pi 2 i.$$
There is some more math and some more subtleties hidden inside. But this should suffice as a first explanation.
Maybe also this might be interesting. You probably know that logarithms "convert" multipication into addition in the sense $\log(ab)=\log(a)+\log(b)$. And you you know that $i\cdot i=-1$. So it would be quite natural to assume that
And thats exactly what you found.