(Note: In an earlier version of this question, the asker referred to this very similar MO question. Apparently neither the explanation in Gompf and Stipsicz (page 163) nor the older MO answer is doing the trick, so I'll give a third explanation.)
We have a surgery diagram in a 3-manifold $M$ with two components $K_1$ and $K_2$. $K_2$ is arbitrary, but $K_1$ is a small linking circle for $K_2$; i.e. $K_1$ bounds a disk $D$ in $M$ which intersects $K_2$ exactly once. The surgery coefficient of $K_2$ is an integer $n$, and the surgery coefficient of $K_1$ is a rational number $r$.
Our goal is to show that if we delete $K_1$ from the surgery diagram and replace the coefficient of $K_2$ with $n-1/r$, the resulting surgery diagram describes the same 3-manifold as the original one.
Let $N_i$ be a regular neighborhood of $K_i$ in $M$. Let $S_i$ denote the surgery solid torus corresponding to $K_i$, so a meridian disk of $T_2$ in an $(n,1)$ curve on the boundary of $N_2$, and a meridian disk of $S_1$ is a $(p,q)$ (where $r=p/q$) on the boundary of $N_1$. Performing the surgery means replacing $N_i$ with $S_i$.
Let $A$ be a small annulus connecting the boundary of $N_1$ to the boundary of $N_2$. We can take $A$ to be a punctured version of the small disk $D$ referred to above. Let $R$ be a thickened neighborhood of $A$.
It is easy to see that $N_1 \cup R \cup N_2$ is a solid torus embedded in $M$. Furthermore, this solid torus is isotopic to $N_2$.
The key claim is that $S_1\cup R \cup S_2$ is also a solid torus. Once we have established this claim it follows that the result of the original surgery diagram can also be described by a single surgery on $K_2$. All that will be left to do is figure out the surgery coefficient.
To see that $S_1\cup R \cup S_2$ is a solid torus, notice that the meridian disk of $S_2$ intersects the annulus $A$ exactly once. (This is where the requirement that $n$ is an integer comes in.)
I'm running out of time, so I'll leave the calculation of the new surgery coefficient ($n-q/p$) as an exercise.