Doubt about equivalence relation
If both $xRy$ and $yRx$ $\forall x,y \in X$, then the indifference relation $I$ is symmetric by definition. Not all symmetric relations are equivalence relations. However, you are given that $R$ is also reflexive and transitive. So we have that $I$ is symmetric, reflexive, and transitive. Therefore $I$, or $\sim$, is an equivalence relation.

You can verify that $\sim$ is an equivalence relation by reading off the definitions of the three properties. $R$, or $\succsim$, is a (weak) preference relation, also known as a total preorder. Total preorders are transitive, complete, and therefore reflexive.

Keep in mind that reflexivity and transitivity does not necessarily imply symmetry: $R = \{(x,x), (y,y), (x,y)\}$ given $X = \{x,y\}$ is clearly reflexive and transitive but not symmetric because $xRy$ but not $yRx$.