Normally, we have some usual space, say, $[0,1]$ or the circle. And define a map $f:X \to X$ which we then iterate and observe tons of properties (it could be mixing, sensitive, etc., a whole zoo of them). So different maps give us different properties on the same space.
In symbolic dynamics, it is just the opposite: same map (the shift), but different spaces. And the resulting systems are very different too.
I guess this 'duality' is the idea behind symbolic dynamics. But somehow it appears mysterious to me.
What makes symbolic dynamics work? What is the idea behind this duality?
I'm not sure I would call these perspectives 'dual' but rather complementary.
Sometimes it's nice if your space is simple and the map is complicated, and sometimes it's nice if your space is a bit strange (like a Cantor set or some fractal attractor), but the map is comparatively simple (a shift map on a sequence space or an inverse limit). It really depends on the questions you're asking, and the tools you have at hand.
In my experience, the former often guides the latter. The shift map is simple (most of the time), so if we can 're-write' (up to some suitable notion of equivalence) a more complicated map on some arbitrary space in terms of a shift on some sequence space, then we can hope to leverage that fact in order to inform us on properties of the seemingly more complicated map.
More precisely, many dynamical properties are preserved (or can be pulled back) along semi-conjugacies (also called factor maps) - surjective maps $f\colon (X,h) \to (X',h')$ such that $f \circ h = h' \circ f$.
For instance, if we want to find a periodic point of $(X',h')$ and we know that $(X,h)$ has a periodic point (because perhaps $(X,h)$ is a sequence space with $h$ the shift, and so it's generally very easy to find periodic points) then we can use the fact that $h^n(x) = x \in X$ to get that $$h'^n(f(x)) = f(h^n(x)) = f(x)$$ and so $f(x)$ is a periodic point for $(X',h')$.
It isn't always the case though that we try to turn topological dynamics into symbolic dynamics - sometimes it's useful to go the other way. For instance in substitutional dynamics, one often wants to be able to rewrite a shift on a sequence space in terms of a rotation on a torus (e.g. see the history of Rauzy fractals and the Pisot conjecture).