To understand the possible spaces of boundary conditions for a TQFT, it is helpful to start in highest dimension.

Suppose you have a $(d+1)$-dimensional nonanomalous TQFT $\mathcal Q$. (The anomalous story is a bit more complicated, because it means that lots of things you might want to write down are only well-defined up to phase factors.) Given a $d$-manifold $M^d$, it makes sense to ask about boundary conditions placed on $M$. Physically, you imagine a cylinder $M \times \mathbb{R}_{\geq 0}$, and ask that the "bulk" $M \times \mathbb{R}_{>0}$ be flooded with your TQFT $\mathcal Q$, and then you try to write down some valid rules for the "boundary values of fields" (at least if your TQFT does come to you with a description in terms of fluctuating quantum fields).

There are a few things you could do:

You could impose some *local* conditions on the values of the fields at the boundary. If you think in terms of classical field theory for a moment, then the "bulk" field theory is some $(d+1)$-dimensional PDE that the fields need to solve (this PDE is called the *equations of motion* EOM), and you could demand that the restrictions of the fields (and their normal derivatives) satisfy some additional $d$-dimensional PDE. You could add some new "boundary fields" that couple to the bulk.

A typical example of this is in the Chern-Simons / WZW
correspondence. Classically, the bulk EOM say that you have a
$G$-bundle with flat connection. This is "topological": the flat connections are flat in any metric. The WZW boundary condition includes a field which is a section of this bundle, and the boundary EOM say that it is a *holomorphic* section. Holomorphicity is something that you can check locally: it is a PDE on the boundary.

You could also impose some *nonlocal* conditions on the boundary. For example, you might ask that some field at some point in $M^d$ have value related in some way to the value of some field at some other far-away point in $M^d$. A more physical example occurs when you ask for the boundary values of a field to satisfy an *integral* equation, rather than a (partial) differential equation. For example, when $d+1=2$, a typical field theory might ask that the field be, say, a holomorphic function $f$. Given a circle $M^1 = S^1$, an interesting boundary value is to ask that the field extend holomorphically over the disk $D^2 \supset S^1$. Of course, this happens exactly when the negative Fourier modes of $f$ vanish. These Fourier modes are something like $\oint_{S^1} z^{-n} f(z) \mathrm{d}z$.

I bring up this distinction to emphasize that when people talk about the collection of "boundary conditions on $M$", they implicitly mean that nonlocal boundary conditions are allowed.

What type of thing is the collection of boundary conditions on $M^d$? Well, because the boundary conditions are allowed to be nonlocal, I can just as well work with the $(0+1)$-dimensional QFT produced by compactifying along $M$, i.e. $\int_{M}\mathcal{Q} := \mathcal{Q}(M \times -)$. But then we're just discussing boundary conditions for a (topological) quantum mechanics model, since that's what $(0+1)$-dimensional QFT is. A "boundary condition" is then just the same as a *state* in the model: it is a thing that you can prepare at time $0$ and then allow to propagate forward in time.

So the punchline is that the collection of boundary conditions on $M^d$ is a Hilbert space: specifically, it is the Hilbert space that $\mathcal{Q}$ assigns to $M^d$.

Ok, with that understood, now we can ask about a $(d-1)$-manifold $M^{d-1}$. If I tell you such a manifold, then you could look at the $d$-manifold $M^{d-1} \times \mathbb{R}$, if you want, and ask about boundary conditions that you can place on $M^{d-1} \times \mathbb{R}$. Well, that's not a very good question because of the noncompactness of $\mathbb{R}$ (you end up not defining elliptic operators). Rather, you could ask about boundary conditions placeable on $M^{d-1} \times \mathbb{R}$ which are *local in the $\mathbb{R}$-direction*. In other words, I will allow you to impose integral equations, but only if the integral is over something that only points in the $M^{d-1}$ direction. I also demand that the boundary conditions be invariant under translation in the $\mathbb{R}$-direction. I might forget to say "translation invariant" below, but I always want it.

Just as before, to understand the structure of this collection, you can compactify along $M$, and consider the now-$(1+1)$-dimensional (T)QFT $\int_{M^{d-1}}\mathcal{Q} := \mathcal{Q}(M^{d-1}\times-)$.

All I want to point out if that this collection is naturally a category. Specifically, suppose you have two local-in-the-$\mathbb{R}$-direction boundary conditions $X$ and $Y$ on $M^{d-1} \times \mathbb{R}$. Place one of them just on $M^{d-1} \times \mathbb{R}_{<0}$ and the other just on $M^{d-1} \times \mathbb{R}_{>0}$. Now you can ask to extend this configuration to something on all of $M^{d-1} \times \mathbb{R}$. Again I will allow you to write down expressions which integrate over $M^{d-1}$, if you want. The extension then looks like an "interface" between the boundary conditions. Anyway, this set of interfaces between $X$ and $Y$ will be $\hom(X,Y)$. If you place an interface at $M \times\{0\}$ and another one at $M \times \{1\}$, and then compactify out that interval $M \times [0,1] \to M$, then you get a (manifestly associative) composition of interfaces.

[Actually, the compactification is subtle, because the interface might secretly depend on a metric in the $\mathbb{R}$-direction. You can do it if the boundary conditions $X,Y$ are "topological in the $\mathbb{R}$-direction", but in general it is harder.]

Ok, fine, so this is why you have a "category of boundary conditions" assigned to $M^{d-1}$. And, just like we had Hilbert spaces in dimension $d$, we have actually *linear* categories in dimension $d-1$. This linearity is essentially the ability to perform "superpositions" in quantum theory. (In unitary quantum theory, it is in fact a Hilbert category.)

Now I can get to your actual question. First, of course, any linear category has an *abelian* group as its $K^0$. Let's understand it. Well, if you do have a local-in-the-$\mathbb{R}$-direction boundary condition on $M^{d-1} \times \mathbb{R}$, then, because it is local and translation invariant, it also defines a (partially local) boundary condition on $M^{d-1} \times S^1$. This assignment takes direct sums to sums.

In other words, there is a canonical map of abelian groups $K^0(\mathcal{Q}(M^{d-1})) \to \mathcal{Q}(M^{d-1} \times S^1)$. The codomain is the Grothendieck group of the category of local-and-translation-invariant-in-the-$\mathbb{R}$-direction boundary conditions on $M^{d-1} \times \mathbb{R}$, and the domain is the Hilbert space of boundary conditions on $M^{d-1} \times S^1$.

**In many cases, this map induces an isomorphism $K^0(\mathcal{Q}(M^{d-1})) \otimes \mathbb{C} \cong \mathcal{Q}(M^{d-1} \times S^1)$.**

This is true, for example, in any (extended) TQFT valued in any member of the Bestiary of 2-vector spaces. You can prove this simply by compactifying down to a $(1+1)$-dimensional TQFT, and then proving a universal statement for $(1+1)$d TQFTs. Cases where it fails have some pathology (nonunitarity or noncompactness or...) that make them less "physical".