Proving regularity via equivalence classes
If you are allowed to use closure properties (as the formulation in the question suggests) then $xw_1\in L_1$ so $w_1$ is a suffix of $L_1$, similarly $w_2$ is a prefix of $L_2$. Taking suffixes and prefixes is a closure property of regular languages, and so is concatenation. That solves the regularity.

At what level do you want/need equivalence classes for your answer?

(added) For language $K$ its set of prefixes is defined as 
$$\mbox{pref}(K) = \{ w \mid \mbox{there exists string $y$ such that } wy\in K  \}$$
precisely as used for $L_2$ in the operation in the question.

Given a finite state automaton for $K$ we get a FSA for $\mbox{pref}(K)$ by making all states final that have a path leading to a final state. 

Also $\mbox{suff}(K) = \{ w \mid \mbox{there exists string $x$ such that } xw\in K  \}$.
Similarly one proves closure under suffix by making all states initial. As that is not commonly allowed for FSA that is solved by adding $\varepsilon$-transitions to all other states.

Your language $L$ based on $L_1,L_2$ equals $\mbox{suff}(L_1)\mbox{pref}(L_2)$, the concatenation of a suffix and a prefix language.

I could not find a reference to a question in this forum that dealt with the prefix/suffix operations, perhaps someone can help. Closure under prefix is a special case of closure under quotient.