The author defines an equivalence relation between sets, i.e. he says that sets are equivalent, if there is a bijective mapping from one to the other. That is a map that uniquely maps all elements of one set to elements on the other.
In short: two sets are equivalent if they have the same number of elements. This is a relation between sets, because it sets two sets into relation to each other, namely, having the same number of elements or not.
Being an equivalence relation "##\sim##" means, it is
- reflexive: ##M \sim M##
- symmetric: ##M\sim N \Longrightarrow N\sim M##
- transitive: ##M\sim N \text{ and }N\sim P \Longrightarrow M\sim P##
Having the same number of elements fulfills these conditions, so we can speak of an equivalence relation here.
The sets ##\{\text{ Soccer }, \text{ Golf }\}## and ##\{\text{ car }, \text{ bike }\}## are obviously not the same, but equivalent, because they have both two elements. We can map ##\text{ Soccer } \longrightarrow \text{ car }## and ##\text{ Golf }\longrightarrow \text{ bike }## and get a bijective map.
If we now compare ##\{\text{ Soccer }, \text{ Golf }\}## with ##\{\text{ Soccer }, \text{ Golf }\}##, then they are clearly equal. But they also have both two elements, which makes them equivalent. E.g. we can map ##\text{ Soccer } \longrightarrow \text{ Golf }## and ##\text{ Golf }\longrightarrow \text{ Soccer }## and get a bijective map. Equality is a special kind of equivalence relation. In our case, it is stronger, because not only the number of elements have to be equal, but the elements themselves have to be as well, in order to have equal sets.