Here is a more detailed account on the history by Dan Voiculescu himself. This is from his article "Background and Outlook" in the Lectures Notes
"Free Probability and Operator Algebras", see
http://www.ems-ph.org/books/book.php?proj_nr=208

Just before starting in this new direction, I had worked with Mihai Pimsner,
computing the K-theory of the reduced $C^*$-algebras of free groups. From the
K-theory work I had acquired a taste for operator algebras associated with free
groups and I became interested in a famous problem about the von Neumann
algebras $L(\mathbb{F}_n)$ generated by the left regular representations of free groups,
which appears in Kadison's Baton-Rouge problem list. The problem, which
may have already been known to Murray and von Neumann, is:*are $L(\mathbb{F}_m)$ and $L(\mathbb{F}_n)$ non-isomorphic if $m \not= n$?*

This is still an open problem. Fortunately, after trying in vain to solve it,
I realized it was time to be more humble and to ask: is there anything I can
do, which may be useful in connection with this problem? Since I had come
across computations of norms and spectra of certain convolution operators on
free groups (i.e., elements of $L(\mathbb{F}_n)$), I thought of finding ways to streamline
some of these computations and perhaps be able to compute more complicated
examples. This, of course, meant computing expectations of powers of such
operators with respect to the von Neumann trace-state $\tau(T) = \langle T e_e,e_e\rangle$, $e_g$
being the canonical basis of the $l^2$-space.

The key remark I made was that if $T_1$, $T_2$ are convolution operators on $\mathbb{F}_m$
and $\mathbb{F}_n$ then the operator on $\mathbb{F}_{m+n} = \mathbb{F}_m \ast \mathbb{F}_n$ which is $T_1 + T_2$, has moments $\tau((T_1 + T_2)^p)$ which depend only on the moments $\tau(T_j^k)$, $j = 1, 2$ , but not
on the actual $T_1$ and $T_2$. This was like the addition of independent random
variables, only classical independence had to be replaced by a notion of free
independence, which led to a free central limit theorem, a free analogue of
the Gaussian functor, free convolution, an abstract existence theorem for one
variable free cumulants, etc.