Why intermediate map in Gysin sequence is multiplication by Euler class

Question

I am reading Bott and Tu, Differential Form in Algebraic Topology. At page 178, they constructed Gysin sequence of Sphere Bundle. I am having trouble understanding the argument, $d_{k+1}$ is multiplication by Euler class. They said, identify an element of $$ E_2^{n-k,k} \cong H^{n-k}(M)\otimes H^{k}(S^k) $$ with $(\pi^{*}(\omega).(-\psi))$, where $\psi$ is the angular form. They said on the level of forms differential on $E_k$ page, $d_k$ is given by the $\delta$ obstruction of extending a d-cycle to a D cycle of length k. I am not able to identify the aforementioned construction with this current scenario. Can you please give me some argument or share some insight on the proof? Thank you.