Write $\tilde{\theta} - \theta = (\tilde{\theta} - \mathbb{E}_\theta[\tilde{\theta}]) + (\mathbb{E}_\theta[\tilde{\theta}] - \theta)$. The first term has mean zero and the second is a constant (the bias). Squaring and taking expectation, the cross term vanishes: \begin{align*} \mathbb{E}_\theta[(\tilde{\theta} - \theta)^2] = \mathbb{E}_\theta[(\tilde{\theta} - \mathbb{E}_\theta[\tilde{\theta}])^2] + (\mathbb{E}_\theta[\tilde{\theta}] - \theta)^2 = \operatorname{Var}_\theta(\tilde{\theta}) + (\operatorname{bias}_\theta(\tilde{\theta}))^2. \end{align*}