I will call $X_n(K)$ the trace of $n$-surgery along $K$, that is a 4-manifold diffeomorphic to the union of $B^4$ and an $n$-framed 2-handle attached along $K \subset S^3 = \partial B^4$.

Call $A \subset S^3 \times I$ the concordance from $K_0$ to $K_1$.
Consider $X_1 := X_n(K_1)$, viewed as $B^4 \cup S^3\times I \cup H$, where $H$ is the 2-handle. For convenience, I will call $C$ the core of $H$. I claim that $X_n(K_0)$ embeds in $X_n(K_1)$ as a regular neighbourhood, that I'll call $X_0$, of $B^4 \cup A \cup C$. This is because a regular neighbourhood of $A \cup C$ (which is a disc) is just a 2-handle $H'$; the framing along which $H'$ is attached is determined by the intersection form, and is bound to be $n$.

Now the second claim is that $W := X_1 \setminus {\rm Int\,} X_0$ is an integral homology cobordism from $Y_0 := S^3_n(K_0)$ to $Y_1 := S^3_n(K_1)$. I will use excision, which tells us that $H_i(W, Y) = H_i(X_1, X_0)$ for each $i$.
Since $H_i(X_0) = H_i(X_1)$ is trivial when $i \neq 0,2$, and since at the level of $H_0$ nothing really happens, we only need to look at $H_2$.

Now, $H_2(X_0)$ is generated by a class represented by a Seifert surface for $K_0$ capped with the core of the 2-handle, that is $A \cup C$. This surface intersects geometrically the co-core $D$ of the 2-handle $H$ of $X_1$ once (since this intersection takes place in $H$, it's exactly $D\cap C$, which is one point), so the generator of $H_2(X_0)\simeq \mathbb Z$ is sent to a generator of $H_2(X_1) \simeq \mathbb Z$. It follows that the relative homology is trivial, as we wanted to show.

As for the addition: any integral homology cobordism invariant now gives a wealth of knot invariants. The Rokhlin invariant, for instance, gives you the concordance invariance of the Arf invariant. I am very partial to Heegaard Floer homology, so correction terms there give you a wealth of concordance invariants. (It should be pointed out that correction terms in Heegaard Floer homology were inspired by work of Frøyshov in Seiberg–Witten theory.)