Show that a language belongs to the polynomial hierarchy
Let me assume your are familiar with the oracle-version of the polynomial hierarchy. Thus, if $L\in \Sigma_k$, then there exists a (non-deterministic polytime) Turing machine $M$ with oracle $\Sigma_{k-1}$.

To show that $L^*$ is also in $\Sigma_k$ we explain how one could build a new non-deterministic polytime Turing machine $M^*$ with oracle $\Sigma_{k-1}$ that accepts $L^*$. The key idea is, that we use the (simulation) of $M$ as a sub-module for $M^*$.

The machine $M^*$ works as follows. It guesses a partition of the input $w$ into some words $u_1u_2\cdots u_k$. Then it runs the simulation for $M$ for every $u_i$. If the simulation verifies that all $u_i\in L$ then $M^*$ accepts, otherwise it rejects. Clearly $M^*$ accepts $L^*$ and uses only the oracle $\Sigma_{k-1}$. What is left to check is if $M^*$ runs in polytime. The running time of $M^*$ is dominated by 
$$t_M(|u_1|)+t_M(|u_2|)+\cdots+ t_M(|u_k|)\le t_M(n)+t_M(n)+\cdots+ t_M(n)\le n \cdot t_M(n),$$ 
for $n=|w|$. Since $t_M(n)$ is a polynomial, we have that $n\cdot  t_M(n)$ is a polynomial as well.