I vaguely remember a 3blue1brown video that explains this really well too. Something to do with thinking about it in terms of rotations on the complex number plane which can land you back on the axis of real numbers. That was the most intuitive thing that I remember seeing
Overly pedantic but I think i^i is actually not well-defined (and in general exponentiation of complex numbers is not well defined) since e^(2 * pi * i) = 1 so for instance we would have i = e^(0.5 * pi * i) = e^(2.5 * pi * i) and then i^i = e ^(-0.5 * pi) = e^(-2.5 * i) but those last two numbers are both real and not equal so complex exponentiation only works when you pick a branch of the complex logarithm. Anyway...
Yes, though regardless of which branch you pick to define i^i, it's value is real.
That proof uses Euler's formula, which is the real brain melter.