Simplying this regular expression
The regular expression on the right consists of the empty word along with all words ending in $10$. All words accepted by the regular expression on the left are of this form. It remains to show that all words accepted by the regular expression on the right are also accepted by the one on the left. This is clear for the empty word, so it remains to show the following:


  Every word ending in $10$ is accepted by $(0^*1^+0)^*$.


Indeed, every word ending in $10$ has the form
$$
0^{a_1} 1^{b_1} 0^{a_2} 1^{b_2} \cdots 0^{a_n} 1^{b_n} 0,
$$
where $b_1,a_2,b_2,\ldots,a_n,b_n \geq 1$. We can decompose this word as
$$
(0^{a_1} 1^{b_1} 0) (0^{a_2-1} 1^{b_2} 0) (0^{a_3-1} 1^{b_3} 0) \cdots (0^{a_{n-1}-1} 1^{b_n} 0) \in (0^*1^+0)^*.
$$