I'd explain the non standard reals as follows.
The language $L$ contains literally all functions ${\mathbb R}^n\to\mathbb R$ and all subsets of ${\mathbb R}^n$. (Recall: symbols can be anything.)
Consider ${\mathbb R}$ as an $L$-structure. Each symbol in $L$ is interpreted in itself.
Now let $^*{\mathbb R}$ be a proper elementarity extension of ${\mathbb R}$.
This makes elementarity and tranfer principle the same thing.
P.S. In the context of NSA (non standard analysis) the real closedness of ${\mathbb R}$ is an infinitesimal detail (excuse the pun). Real closedness is concerned only with the order-algebraic part of ${\mathbb R}$. This is just a tiny fraction of the language you use in NSA.