This is just an informal way to make you understand the implied definition. "merging" here means that the two copies of $1$ are sent to the same thing.
Formally, the map $X+1+1\to X+1$ is defined : on $X$ by the inclusion $X\to X+1$, on the first $1$ by the inclusion $1\to X+1$ and on the second $1$ by the same inclusion $1\to X+1$. Since $X+1+1$ is a coproduct of these three things, this map is well-defined and it has "identified" or "merged" the two copies of $1$