Roughly, we are told that it has to be one or the other that is true; then we are told that it is not the one that is true; so we infer that it has to be the other that is true. The reason this is called "disjunctive syllogism" is that, first, it is a syllogism--a three-step argument--and second, it contains a disjunction, which means simply an "or" statement. "Either P or Q" is a disjunction; P and Q are called the statement's disjuncts.
Here is an example:
Either I will choose soup or I will choose salad.
I will not choose soup.
Therefore, I will choose salad.
Here is another example:
Either the Browns win or the Bengals win.
The Browns do not win.
Therefore, the Bengals win.
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