Is the language of binary strings that contain a substring of the form $ww$, where $w \in (0+1)(0+1)^*$ regular?
Your language is regular.
Let's see which words do not belong to $L$. If a word contains $00$ or $11$ as a substring that it does belong to $L$, hence any word not in $L$ must have alternating $0$s and $1$s. If such a word has length at least $4$ then it must start with $0101$ or $1010$, and so again it is in $L$. We are left with the following words, which comprise $\overline{L}$:
$$ \epsilon, 0, 1, 01, 10, 010, 101. $$
Since $\overline{L}$ is finite, it is regular, hence so is its complement $L$.
What this argument shows is that any word in $L$ contains one of $00,11,0101,1010$ as a substring. This explains the given regular expression.