The text can be found here: https://www.maths.ed.ac.uk/~v1ranick/papers/atiyahk.pdf
I am having two struggles on the proof of Lemma 1.4.9. First, why is $C^+(X) \cap C^-(X) = X$? To me, it would appear that it should be $\{1/2\} \cup X/(\{0\} \cup \{1\}) \cup X$. Secondly, I am confused why $E|C^+$ and $E|C^-$ are trivial, or how to prove that $E$ is a vector bundle. Any help would be greatly appreciated!
$S(X)$ is the suspension of $X$, where $X$ is identified with the middle $\{\frac{1}{2}\}\times X$, and $C^+(X)$ is the top part, i.e. $[\frac{1}{2}, 1]\times X/ (\{1\}\times X)$, and $C^-(X)$ is the bottom part.
$E|_{C^+}$ and $E|_{C^-}$ are trivial, because $C^{\pm}$ are contractible (Lemma 1.4.4 (2)).
As for how to prove $E$ is a vector bundle, the proof starts with $E$ being a vector bundle on $S(X)$, so nothing to be proved here.