In Atiyah, Bott, and Shapiro's Clifford Modules, Theorem 12.3, they prove that a Spin$(k)$-structure (resp. Spin$^c(2k)$)-structure gives a KO(K)-orientation on the associated vector bundle of rank $k$ ($2k$). In the comment directly after the theorem, they state the following:
Remark. It is easy to see, by considering the first differentials in the spectral sequence, that the existence of a Spin (Spin$^c$)-structure is necessary for KO(K)-orientability. Theorem (12.3) shows that these conditions are also sufficient.
It is not easy for me to see this. I assume the spectral sequence in question is the Atiyah-Hirzebruch spectral sequence, but which bundle is it applied to? One can apply it to the vector bundle itself, but I don't see how this gives something related to Spin-structure. Can anyone elaborate a bit more on their argument?