Why is it always the case that, given an equivalence relation R on a set X and a $x \in X$, that x is equivalent to itself?
According to my book, this is due to the reflexivity of $R$ and consequently $x \in [x]_R$, but isn't an equivalence relation supposed to be reflexive, symmetric and transitive - not just reflexive?
I've cross-checked this with some other books, and they all say the same thing. I don't get it, how can $x \in [x]$ if x only satisfies reflexivity?
You're mixed up about something, and it's really not clear what.
Let $R$ be an equivalence relation.
$R$ is reflexive. Yes, $R$ is also symmetric and transitive, but that doesn't change the fact that $R$ is reflexive.
We can prove $xRx$ using only the fact that $R$ is reflexive. That doesn't prevent $R$ from having additional properties beyond being reflexive.
The above paragraph also applies to showing $x \in [x]_R$. (depending on the definition of $[x]_R$, I suppose)
Reflexivity is a property of relations, not elements. It doesn't make sense to say that $x$ is reflexive.
If having the facts of the matter laid out doesn't help, you'll probably have to give a better explanation of how and why you've arrived at the the things you think are true and at the questions you have and why you're confused about them.