It is just a direct application of the definitions, but you can simplify the computation of the squared supercovariant derivatives a little bit by introducing a coordinate change in the superspace, as I will describe below.
First recall the definitions. Let f(x,θ,ˉθ) be a function on superspace. One can always expand it in terms of θ,ˉθ as f(x,θ,ˉθ)=c1(x)+θc2(x)+ˉθc3(x)+θ2ˉθ2c4(x).
By definition, the integral over d4θ=d2θd2ˉθ is ∫f(x,θ,ˉθ)d4θ=c4(x).
By a direct computation, ∫f(x,θ,ˉθ)d4θ=116[∂∂θα∂∂θα∂∂ˉθ˙α∂∂ˉθ˙αf(x,θ,ˉθ)]|θ=ˉθ=0.
Introducing a simple coordinate change, yμ=xμ+iθ˙ασμα˙αˉθ˙α, we can rewrite the fermionic derivatives as: Dα=∂∂θα+2iσμα˙αˉθ˙α∂∂yμ,ˉD˙α=−∂∂ˉθ˙α.
This transformation is usually employed when dealing with chiral and anti-chiral superfields, but here it simplifies the computation since ˉD2=ˉD˙αˉD˙α trivially equals (∂/∂ˉθ˙α)(∂/∂ˉθ˙α), while DαDαf(x,θ,ˉθ)=(∂∂θα+2iσμα˙αˉθ˙α∂∂yμ)2f(x(y),θ,ˉθ)=∂∂θα∂∂θαf(x(y),θ,ˉθ)+O(θ,ˉθ),
where O(θ,ˉθ) are first-order terms in the fermionic variables, which vanishes when we apply [...]|θ=ˉθ=0. So we have derived that:
116[D2ˉD2f(x,θ,ˉθ)]|θ=ˉθ=0=∫f(x,θ,ˉθ)d4θ.
Your particular result follows once you expand the superfield functional K[Ψ,ˉΨ] in term of the superfields and apply the above identity.