I think that the following story is quite illuminating for introduction. Naturally, one can/should adapt/change it to better fit the audience, I just wanted to sketch the general idea.

> Suppose you came here by bus. Which bus was that? You say it was the _42_, which is the line that goes from the main station to the university. However, was it really the _42_? Is there only a single bus carrying this number?
>
> No, there are multiple buses marked _42_, but you don't care which one you rode, because they all take the same route. You _abstracted_ the differences between particular cars, because they were _equivalent_ for your purposes. You have just used the concept of _equivalence classes_.

After that, I try to explain the intuition using graphs (informally), like [here](http://math.stackexchange.com/a/740084/26306).

I hope this helps $\ddot\smile$