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I'm talking about structuralism in an anthropological and philosophical context, that is, the notion that human behaviour and perception are determined by structures (linguistic, cultural...) that are beyond human agency. The position is worked to its extreme conclusions (in post-structuralism) when it 1. reduces all human behaviour to such determination, and 2. that human agency itself is an illusion.

It is clear how Freud and Marx pour into this stream. Both highlighted how human behaviour is fundamentally determined by non-obvious and hidden psychological and social factors. However, the rise of structuralism in both anthropology and philosophy is accredited to Ferdinand de Saussure.

What specifically in Saussure's thought gave rise to structuralist anthropology and philosophy in the sense understood above?

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To say that human agency itself is an illusion is a vey strong statement, Bretanos notion of intentionality and Husserls phenomenology take a point of view that opposes this by looking at the world from the direct view of consciousness; Heidegger,too, probably fits along this line too as he considers Being in Time.

Structuralism has two roots in modern philosophy, the more well-known has roots, as you recalled, in Saussures structural linguistics; the other root is in Bourbaki, the structuralism of modern mathematics whose modern avatar is category theory; notably only the first has had an impact on Philosophy through Saussures structuralism being taken up by Levi-Strauss in his Anthropology; Bourbakis structuralism has had a wide impact on mathematics but is struggling to make its voice heard elsewhere; even in such a close and neighbourly discipline such as Physics. Mathematics has had a wide impact on the flavour of philosophy of the analytic kind - through the foundational study of mathematics and logic - which has developed in Wittgensteins analytics philosophy of language (predominantly of the propositional kind - which is why it slips past poets and prose sylists).

Structuralism in linguistics avoids all questions of semantics that is meaning in favour of a purely structural view of language; that is the parts of language and how they inter-relate; that is:

The foundation of structural linguistics is a "sign," which in turn has two components: a "signified" is an idea or concept, while the "signifier" is a means of expressing the signified. The "sign" is thus the combined association of signifier and signified.

This amplifies and analyses the older idea of a word 'dog' (the sign or signifier) and the dog itself (the signified); but notably now a picture of a dog is also signifier; and similar things can be said for the visual grammar and iconography of advertising, cinema and architecture.

Signs can be defined only by being placed in contrast with other signs, which forms the basis of what later became the paradigmatic dimension of semiotic organization (i.e., collections of terms/entities that stand in opposition). This idea contrasted drastically with the idea that signs can be examined in isolation from a language and stressed Saussure's point that linguistics must treat language synchronically.

Here two things are being said, that the sign 'dog' is understood as different from the sign 'cat'; and that to understand the sign 'dog' one must also understand the entire language; this, at first, appears odd - do we not understand the sign 'chien' as the french word for the sign 'dog'? And this without understanding the entire French Language? We do, of course; but one must understand that for humans that to know one language, is in some sense to know them all; for they all have structural similarities (this is Chomskys thesis).

They add:

Saussure set out to model language in purely linguistic terms, free of psychology, sociology, or anthropology. That is, Saussure was trying precisely not to say what goes on in your or my mind when we understand a word or make up a sentence. [...] Saussure was trying to de-psychologize linguistics

This doesn't mean that linguistics can be depsychologised; but that we ask what can we know about language if we are prohibited from utilising our knowledge of sociology and psychology; in this sense it has some commonality with Skinners Behaviouralism; and the object-orientation of Category Theory.

Levi-strauss applied this platonic theory to Anthropology; to understanding the structures of myth without understanding what these myths signified to the clan or tribe. Thus its myths are signs, and made up of further signs.

addendum

One has to be deeply read in either linguistics or anthropology, I suspect, for the structural approach to begin to make sense; I can't speak for structuralism in the afore-mentioned subject, but certainly its true that structuralism in mathematics, which I do know something about, only begins to make sense when one has completed an undergraduate course - at least for a typical student - there are of course atypical ones, which tends to happen more with mathematics because of its subject matter than the humanities.

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