For the sake of simplicity let us rename the two languages $L_1$ and $L_2$ as $A$ and $B$ respectively.
Let $M_A(Q_A,\Sigma,\delta_A,q_{0A},F_A)$ and $M_B(Q_B,\Sigma,\delta_B,q_{0B},F_B)$ be two DFA accepting the language $A$ and $B$ respectively.
To prove that the language generated by $shuffle(A,B)$ is regular, we define a NFA $M(Q,\Sigma,\delta,q_{0},F)$ which accept all strings of language $shuffle(A,B)$.
Let, $Q = Q_A \times Q_B$, $q_0 = (q_{0A}, q_{0B})$ and $F = F_A \times F_B$. Then we define the transition function of $M$ as follows
$$\delta((q_a, q_b), s) = {(\delta_A(q_a,s),q_b)} \cup {(q_a,\delta_A(q_b,s))}$$
This machine will accept exactly those string which are in $shuffle(A,B)$ and equivalently $shuffle(L_1,L_2)$ and thus completes the proof that the language $shuffle(L_1,L_2)$ is regular.
HTH