Basic idea is to give a mathematical characterization of a ‘world’ in just enough detail to assign a meaning to every expression. Exactly what counts as ‘just enough’ depends on the language.
Key point is that MT makes only the assumptions about The World that are needed to determine truthvalues, and they must be expressible mathematically, ie ‘structurally’. Model theory is metaphysically neutral
radically agnostic. Eg first-order MT makes no assumptions about what is in the universe.