The "Natural" in Natural Semantic Metalanguage is intended to contrast with other semantic metalanguages which use non-linguistic symbols and syntax.
Here's an example, which apparently is describing the semantics of have to:
[[have to ϕ]]w,f,g = 1 iff for all v ∈ f(w) such that there is no v' ∈ f(w) such that g(w)(v',v),[[ϕ]]v,f,g = 1
In contrast, a NSM explication of have to is:
[A] Someone X has to do something (VP):
a. it is like this:
b. this someone X can’t not do something (VP)
c. someone can know this
So rather than using mathematical/logical symbols (∀, ¬, ⊃, ∄), symbols taken from natural language are used (ALL, NOT, SOME, IF). And rather than using the syntax of functions or following the order of operations, the syntax of natural language is used, following (in English for example) the normal order of SVO, adjective-noun etc.
Whether this is a good idea, and whether it's successful is definitely still up for debate, but that's what the "Natural" means.