After formally stating the central limit theorem my statistics textbook says this:
Interpretation: Probability statements about the sample mean $\overline{X}_n$ can be approximated using a Normal distribution. It's the probability statements that we are approximating, not the random variable itself.
What distinction are they trying to make here? What would it mean to approximate a random variable other than to approximate probability statements about it?
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It is quite common for two different random variables $Z_1$ and $Z_2$ to have the same probability distribution, (and therefore probabilistically speaking be representing the same object), but be emerging from two independent systems, and therefore act completely independent of one another.
So although they are not the same function (random variables are actually functions that assign values to different outcomes in a corresponding sample space), when you examine both of them, they have the same probability distribution and therefore same probabilistic properties. This is what is being approximated in your question. And unfortunately as Michael said in his comment, a book in statistics isn't supposed to expect you to make such a subtle distinction without knowing beforehand how to look at random variables as measurable functions.
In summary: Although $\bar{X}_n$ doesn't resemble a Normal random variable, it's probability distribution does resemble the one that a normal random variable would have
Under certain conditions, the probability $P(\bar{X}_n \in [-1, 1])$ (for example), which is a non-random number, can be approximated by $P(Y \in [-1, 1])$ for some normal random variable $Y$ with appropriate mean and variance, using the central limit theorem. To emphasize, you are approximating the probability $P(\bar{X}_n \in [-1, 1])$, and not the random variable $\bar{X}_n$ itself: you are not saying something is close to $\bar{X}_n$.