equivalency of partition & equivalency relation's definitions

To get you started with the proof: Suppose you are given a partition $S= \{S_i\}$ and the relation $R$ defined as in your question. You have to show this is an equivalence relation, so you need to verify reflexivity, transitivity and symmetry. I'll show transitivity and let you try to do the rest:

So assume $a\sim b $ and $b\sim c$. This means by definition of $R$ that there is $S_1 $ such that $a\in S_1$ and $b\in S_1$, and (second relation) $S_2$ such that $b\in S_2$ and $c\in S_2$. Since we have a partition, $b\in S_1$ and $b\in S_2$ implies $S_1 = S_2$. So we have $a\sim c$.

Basically you have to recall the definitions and verify them...of course what I have shown is only part of the first direction.

As for the example: you have a partition, so it defines an equivalence relation according to the formula in your question. This just means that you have the relations $1\sim 3, 2\sim 5$, the associated symmetric relations and the trivial ones $a\sim a$ for each $a$. You don't have to find them, they are defined by the partition.

(This was the very first exercise I had to solve when I studied math some 30 years ago, so I understand your confusion...)