Why the conditional expectation is better
Let U be unbiased for \theta and let T be sufficient. In one sentence each, say why W(T)=\mathbb{E}(U\mid T) is a statistic, why it is unbiased, and why \operatorname{Var} W\le\operatorname{Var} U.
Worked answer
W(T) is a statistic because sufficiency makes the conditional distribution of the sample given T free of \theta, so the conditional expectation does not depend on \theta. It is unbiased because \mathbb{E}\{W(T)\}=\mathbb{E}(U)=\theta by the tower property. Its variance is smaller because \operatorname{Var}(U)=\operatorname{Var}\{W(T)\}+\mathbb{E}\{\operatorname{Var}(U\mid T)\} and the second term is nonnegative.