Zermelo–Fraenkel set theory the natural numbers defines $1$ as $1 = \{\{\}\}$ but this does not seem right If 1 can be defined as the set that contains only the empty set then what of sets which contain one thing such as the set of people who are me.
number 1 does not just mean $1$ nothing, it means $1$ something. The definition does not seem to capture what we mean by numbers when we use them in our everyday lives.
I think I may be missing something in the explanations I have read.
Can anyone put me right on this.

This definition has almost nothing to do with the "meaning" of $1$.
This is a familiar concept, just in an unfamiliar context. What is "sun"? Is it a gigantic ball of burning gas? Nope; it is simply three curvy lines drawn next to each other. It is a word that refers to a thing, but is not the thing itself.
The definition you cite is almost completely analogous: it's just that the purpose is to write one as a set in a set-theoretic universe so that we can treat it set-theoretically, rather than writing it as a word in the English language that we can treat linguistically.
But it does turn out that in this writing scheme, the set chosen does have some relation to the meaning: we have encoded one as a set whose cardinality is one (i.e. as a set with one element).
This is sort of like the word "sesquipedalian" which happens to be an instance of the concept it refers to. Although in this instance it is more practical, since it allows the things that relate to the meaning of $1$ to translate more directly to how we have written it: e.g. a set has one element if and only if there is a bijection from that set to $1$. (that is, to the set we have chosen to write $1$ as)
Usually, we would write this definition instead as $1 = \{ 0 \}$, given that we have previously defined $0 = \{ \}$.
