A surgery on a link in $S^3$ can be regared as 2-hadnle attachements to $D^4$ (resulting 4-manifold $W$ whose boundary is the result of the surgery).
I would like to know how the first Kirby move (adding the trivial knots with framing $\pm 1$) corresponds to the connected sum of $\mathbb{C}P^2$ or $\overline{\mathbb{C}P^2}$ to the 4-manifold $W$.
I am not familiar with the topology of $\mathbb{C}P^2$.
I will just deal with the case of adding in unknots with framing $+1$. Everything is entirely analogous in the $-1$ case.
$\Bbb CP^2-\{pt\}$ is naturally an open $D^2$-bundle over $S^2=\Bbb CP^1$. Consider $p:\Bbb CP^2-[0:0:1] \rightarrow \{z=0\} $ with $p([x:y:z]) = [x:y:0]$. In words, $p([x:y:z])$ is the unique point on $\{z=0\}$ intersecting the line between $[x:y:z]$ and $[0:0:1]$. Similarly, one can fiber the complement of an open ball in $\Bbb CP^2$ as a closed $D^2$-bundle over $S^2$. In particular notice $\partial (\Bbb CP^2-B^4)=S^3$
$D^2$-bundles over $S^2$ are classified by their Euler number (the self-intersection of their zero section). As the zero section of our above bundle is a line in $\Bbb CP^2$, it has self intersection $+1$ (as any two lines in $\Bbb CP^2$ intersect in one point with positive intersection).
There is an alternative way to realize a $D^2$-bundle over $S^2$ with Euler number $n$ as the $2$-handlebody given with one $0$-handle and one $2$-handle attached along the unknot with framing coefficient $n$. For $n=+1$, the classification of $D^2$-bundles implies this is bundle-isomorphic to our above bundle, and in particular $\Bbb CP^2-B^4$ is diffeomorphic to this handlebody.
Now if we have some arbitrary $2$-handlebody $X$, and we boundary sum it with $\Bbb CP^2-B^4$ the new boundary will be $\partial X \#S^3\cong \partial X$. The Kirby diagram for this boundary sum is just the diagram of $X$ next to the diagram of $\Bbb CP^2-B^4$ (i.e. adding a $+1$-framed unknot to the diagram of $X$). So we have realized our desired Kirby move.
See Gompf-Stipsicz 4-Manifolds and Kirby Calculus for more details.