Measuring the hardness of SAT instances Lemma
First, $[a\to 1]\Gamma = (1)(1)(c)(c')$. A refutation of this consists of resolving $c$ and $c'$. Following the proof, we reinstate $a'$ to get the CNF $(1)(1)(a'+c)(a'+c')$, and then resolve $a'+c$ and $a'+c'$ to get $a'$.

Similarly, $[a\to 0]\Gamma = (b)(b')(1)(1)$, which can be refuted by resolving $b$ and $b'$. Reinstating $a$, we get $(a+b)(a+b')(1)(1)$, and resolving $a+b$ and $a+b'$ we get $a$.

Note that your notation, which is different from the paper's, is confusing you. For example, $x'$ is not the negation of $x$ but rather a new variable symbol.