Would someone give an easy explanation?
(Yet more on this here).
$1=0+1$, $1^2=0\times10^m+1$ seems to fit the definition as given at the OEIS reference.
Unfortunately, my answer is anticlimactic. I happen to know wikipedia's definition (linked in the question, but I reproduce the definition here)
So $A$ can be $0$. Thus $1^2 = 1 = 0* 10^1 + 1$, and we see that it's a Kaprekar number.
And - Gerry posted his answer just before me (I refreshed, and it's there)! But I wrote this too, so I'll keep it -