Here's perhaps a way to understand the general case that Neil mentioned, and a way to apply it to your question at the end. 

Suppose you have a category $C$. Then there's a free symmetric monoidal category on $C$, $FC$, which Neil described in his answer. 

Now if you're looking at $F^0(C,Set)$ with respect to some $c\in C$, the category that you get only depends on $BEnd(c)\subset C$, the full subcategory on $c$ (which is a one-object category, so you can see it as a monoid somehow), and in fact its core-groupoid (which is the only thing that $K$-theory depends on) is exactly $F(BEnd(c)^\times)$, i.e. finite free $End(c)^\times$-sets, where $End(c)^\times$ is the subgroup of invertible elements of $End(c)$. 

The reason for this is that left Kan extension along $\{c\}\to C$ factors as left Kan extension along $\{c\}\to BEnd(c)$, and then left Kan extension along $BEnd(c)\to C$, but the latter is a full subcategory inclusion, therefore left Kan extension along it is one as well, so if you're looking at the category of people that are left Kan extended from $\{c\}$ (from a finite set I would assume, to avoid Eilenberg swindle type phenomena), you might as well look at the same subcategory, but in $Fun(BEnd(c),Set)$, so you might as well assume $C = BEnd(c)$. Then you can notice that a good description of this category is just finite copies of $End(c)$ with its $End(c)$-action, and the core-groupoid of that will just be the same thing but for $End(c)^\times$


Therefore you get exactly the same situation as for a group : $K(Fun^0(C,Set)) = \Sigma^\infty_+(B(End(c)^\times))$ where my $B$ is your $|Nerve(-)|$. 

Now the reason for that thing is that when $C$ is a groupoid (e.g. $B(End(c)^\times)$), (the nerve of) $FC$ happens to also be the free symmetric monoidal $\infty$-groupoid on $C$, i.e. the free $E_\infty$-space on $|Nerve(C)|$; hence if you apply group completion to it, you get the free grouplike $E_\infty$-space on it, in other words (up to delooping) the free connective spectrum on it. But that is precisely $\Sigma^\infty_+|Nerve(C)|$. As Neil pointed out, this gets you the tom Dieck splitting, and in fact with the right setup for $G$-spectra it can give you an "equivariant Barratt-Priddy-Quillen" theorem. 

Certainly this idea is present in Segal's *Categories and cohomology theories* - where he gives a proof "along those lines" of the Barratt-Priddy-Quillen theorem. A modern reference where this idea is explicitly used that way to prove the tom Dieck splitting is Barwick's *Spectral Mackey functors and equivariant algebraic $K$-theory (I)*, specifically theorem A.9. His argument can be generalized to describe the free $E_\infty$-space on a $1$-groupoid.