TAPL: Explanation and example(s) for satisfied
One way of reading the definition of satisfied that might help is to change


  A rule is satisfied by a relation if, for each instance of the rule, either the conclusion is in the relation or one of the premises is not.  


to the logically equivalent statement


  A rule is satisfied by a relation if, for each instance of the rule, if all the premises are in the relation, then the conclusion is in the relation.


Thus if  the rule says, for instance,

$$
\dfrac{A \to B \quad B\to C}
{A\to C}
$$

Then binary relation $R$ satisfies this rule whenever for each instance of the rule, such as

$$
\dfrac{a \to b \quad b\to c}
{a\to c}
$$

if $(a,b)\in R$ and $(b,c)\in R$, then $(a,c)\in R$.

Another way of stating this is that $R$ is closed by the rule (though this probably doesn't make it clearer).

Now let's return to the original statement. What


  A rule is satisfied by a relation if, for each instance of the rule, either the conclusion is in the relation or one of the premises is not.  


is saying is that is it okay to have $(a,c)$ in the relation without the premises, but if all of the premises are in the relation, then  $(a,c)$ must be. That is, it is okay to have not all of the premises in the relation, without imposing any additional constraint.

Personally, I think that thinking about this definition in terms of an implication is simpler, so I help this explanation helps.