I was Thurston's (undergraduate) student in the mid 1980s, when he was thinking about linkages. Here's how Thurston explained the proof to me.

By a theorem of Nash (see https://en.wikipedia.org/wiki/Nash_functions), any smooth manifold is diffeomorphic to a solution space of a set of real polynomial equations. So now all one needs to do is devise planar linkages which implement addition and multiplication of real numbers, and hook them together in a way that mirrors the algebraic equations from Nash's theorem.

I never worked through the details of the above sketch myself.

I was Thurston's (undergraduate) student in the mid 1980s, when he was thinking about linkages. Here's how Thurston explained the proof to me.

By a theorem of Nash (see https://en.wikipedia.org/wiki/Nash_functions), any smooth manifold is diffeomorphic to a solution space of a set of real polynomial equations. So now all one needs to do is devise planar linkages which implement addition and multiplication of real numbers, and hook them together in a way that mirrors the algebraic equations from Nash's theorem.

I never worked through the details of the above sketch myself.

[Added: Igor Rivin's answer overlaps with mine, and has additional details.]