The word "equivalent" is built from "equal" and "value" — well, actually from the Latin words aequus (equal) and valere (to be worth). So two elements are equivalent if they are in a sense equal-valued, or interchangeable. Often this is indeed used literally, by defining some quantities as equivalence classes; for example, the fraction $\frac42$ has the same value as (is equivalent to) the fraction $\frac21$.
Now the properties of an equivalence relation can be directly obtained from that interpretation of the word:
- Clearly some object has to have the same value as itself; thus any equivalence relation fulfils $a\sim a$ for all $a$ (reflexivity).
- Just as clearly, if $a$ has the same value as $b$, then $b$ has the same value as $a$. That is $a\sim b\implies b\sim a$ (symmetry).
- And of course, if $a$ has the same value as $b$, and $b$ has the same value as $c$, then $a$ has the same value as $c$, that is $a\sim b\land b\sim c\implies a\sim c$ (transitivity).
You may note that in the list above, I've ultimately used the exact same properties of equality. That is not accidental; equality is in a way the prototype of an equivalence relation.
Or in short, an equivalence relation describes a more general notion of "sameness".