All strings in which every substring 000 appears after every 1
The formulation

every substring 000 appears after every 1

is extremely ambiguous. However, given the proposed solution, here is how we are supposed to interpret it. Let $w$ and $x = x_0 \ldots x_{m-1}$ be words. We say that $x$ occurs in $w$ at position $i$ if $$w_i w_{i+1} \ldots w_{i+m-1} = x_0 x_1 \ldots x_{m-1}.$$
The language in question consists of all words $w$ such that if $1$ occurs in $w$ at position $i$ and $000$ occurs in $w$ at position $j$, then $i &lt; j$. This means that after the first time that $000$ occurs, the word consists only of $0$s. Therefore we can break words in the language into two parts: the first part doesn't contain $000$ at all, and the second part, which is optional, contains $000$ followed by any number of $0$s.