Proving that context-free languages are closed under inserting symbols
Using closure under GSM mappings

Whenever possible, it may be faster to prove closure properties from
other known closure properties, That is why there has been a
systematic study of closure properties (see AFL, full AFL, trios,
...). This is somewhat developed in the Hopcroft-Ullman textbook,
ed. 1978).

For this problem, the insertion of a single character from the alphabet in the strings of a
language can be trivially performed by application of a
GSM-mapping. Since CF languages (and many others) are closed under
GSM-mapping, they are closed under the $+$ operator.

If you do not know about GSM mappings, you can do it with closure
under intersection with regular sets, homomorphism and inverse
homomorphism. That is actually how closure under GSM mapping is
proved.

Assuming your original alphabet is called $\Sigma$, you create a new
alphabet $\hat\Sigma=\{\hat a \mid a\in \Sigma\}$, i.e. a new set of
symbols in 1-1 correspondance with $\Sigma$.  Using the inverse of a
homomorphism (that simply erases $\hat\Sigma$), you insert an arbitrary
number of $\hat\Sigma$ symbols in the string.  Then you intersect the
result with the regular set $\{w\in (\Sigma\cup\hat\Sigma)^*\mid
|w|_{\hat\Sigma}=1\}$, so as to allow the insertion of only one
symbol from $\hat\Sigma$. Then you remove the hat with a homomorphism $\phi$ such
that $\forall \hat a \in \hat\Sigma,\, \phi(\hat a)= a$.