The standard presentation of hyperreals is difficult to understand. One typically motivates the hyperreals by a desire to have a consistent theory of infinitesimal elements, and then introduces the idea of the transfer principle. This is difficult to understand intuitively because most people don't have intuition about which statements can and cannot be expressed as first-order statements.
One might instead wonder whether we can say something like "a hyperreal field is a minimal field that contains $\mathbb{R}$ and an infinitesimal element" but this is false as discussed here. Any actual hyperreal field contains a lot of elements that $\mathbb{R}(\epsilon)$ doesn't. Still, it seems that the motivation behind hyperreal fields is to have a minimal field that contains $\mathbb{R}$, an infinitesimal element, and enough other elements so that we can "do calculus" on this field.
Is there a formalization of this concept that's more intuitive, such as "a hyperreal field is a minimal field $F$ that contains $\mathbb{R}$ and at least one infinitesimal element and supports the following operations: ... (e.g., extensions of certain real-valued functions to $F$) such that the following identities hold ... (e.g., a list of identities that are similar to the ones obeyed by those functions on $\mathbb{R}$)"?