Here is a short video (by someone else) explaining the general outlines of the argument:
Here is a link to the paper by Harry Binswanger:
A Refutation of Determinism
(1) The argument shows that volition is a prerequisite of all knowledge and that the denial of volitional choice implies self-contradiction, thus establishing volitional choice as an axiom of epistemology. Strictly speaking, I think, by Leonard Peikoff's definitions, this is not a proof, but an axiomatic validation. It establishes the axiom as absolutely and unquestionably true, but does not involve the specific process of logical proof of the positive proposition.