The K$... variable is a parameter. It is likely generated internally by Module[], and it shouldn't be left in the answer. I'm not sure why you're seeing it, but it is helpful. If we replace it by k, we can construct the following parametrized family of solutions:
mySol // ClearAll;
mySol[k_?NumericQ] :=
First@SolveValues[{ComplexExpand[
Abs[x + y*I] == Abs[(x + y*I)^(x + y*I)],
TargetFunctions -> {Re, Im}],
0 <= x <= E^k && y == Sqrt[E^(2 k) - x^2]}, {x, y}];
Then
Table[N@mySol[k], {k, 0, 5, 1/4}] // ListLinePlot
