Common assumptions are that $$ \textrm{Cov}(\mathbf{u}, \mathbf{e}) = \mathbf{0} $$ $$ \textrm{Cov}(\mathbf{e}) = \sigma^2_e \mathbf{I}. $$
Let $i \neq i'$.
On the one hand, we have $$\begin{align*} \textrm{Var}(y_{ij}) & = \textrm{Var}(\beta_0 + u_j + e_{ij}) \\ & = \textrm{Var}(u_j + e_{ij}) \\ & = \textrm{Var}(u_j) + \textrm{Var}(e_{ij}) + 2 \textrm{Cov}(u_j, e_{ij})\\ & = \sigma^2_u + \sigma^2_e. \end{align*}$$
On the other hand, we have $$\begin{align*} \textrm{Cov}(y_{ij}, y_{i'j}) & = \textrm{Cov}(\beta_0 + u_j + e_{ij}, \beta_0 + u_j + e_{i'j}) \\ & = \textrm{Cov}(u_j + e_{ij}, u_j + e_{i'j}) \\ & = \textrm{Cov}(u_j, u_j) + \textrm{Cov}(u_j, e_{i'j}) + \textrm{Cov}(e_{ij}, u_j) + \textrm{Cov}(e_{ij}, e_{i'j}) \\ & = \sigma^2_u. \end{align*}$$
Hence $$\begin{align*} \textrm{Cor}(y_{ij}, y_{i'j}) & = \frac{\textrm{Cov}(y_{ij}, y_{i'j})}{\sqrt{\textrm{Var}(y_{ij})}\sqrt{\textrm{Var}(y_{i'j})}} \\ & = \frac{\sigma^2_u}{\sqrt{\sigma^2_u + \sigma^2_e} \sqrt{\sigma^2_u + \sigma^2_e}} \\ & = \frac{\sigma^2_u}{\sigma^2_u + \sigma^2_e}. \end{align*}$$
The latter is the correlation between measurement $y_{ij}$ and measurement $y_{i'j}$ ($i \neq i'$), i.e., the correlation between "any two responses having the same $j$".
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