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Take a collar ${\partial }_{i}X\times [0,\epsilon ]$. In the quotient, the two annuli $A_1,A_2$ are folded together by $f$, while their common boundary curves ${\beta }_1,{\beta }_2$ become the boundary of a meridional disk. Because $f({\alpha }_1)={\alpha }_2$, the two halves of $\alpha$ join across this disk and bound it. The quotient of the collar is therefore a solid torus whose meridian is attached along $\alpha$; outside the collar nothing changes. Hence the annulus-quotient model of Dehn filling gives

Let $Y_1^{\prime }=X_1({\lambda }_1)$ and orient its filling core $K_1^{\prime }$ so that its longitude is ${\mu }_1$. Then zero surgery on $K_1\#K_2$ is the splice of $(Y_1^{\prime },K_1^{\prime })$ and $(Y_2,K_2)$. Cutting the connected-sum exterior along its decomposing annulus and applying the annulus-quotient model of Dehn filling identifies ${\lambda }_1$ with ${\lambda }_2$ and ${\mu }_1$ with ${\mu }_2$, which is exactly the splice gluing.