What's the easiest way to see this? The only thing I could think of was to try to patch together trivializations. I couldn't find a way to make that work. Thank you!
edit: For the record, here's why I asked about this special case of the more general result about fiber bundles over contractible spaces.
In the much beloved book by Bott and Tu, it's claimed that the Leray Hirsch theorem can be proven in the same way the Kunneth theorem is proven: induct on the size of a finite good cover for the base space, then apply the Mayer Vietoris sequence and the Poincare lemma for the induction step. It's assumed that there exists a finite good cover for the base space, but it's not assumed that this cover is a refinement of the cover of the base space, which gives the local trivializations of the fiber bundle. Therefore, to apply the Poincare lemma in the induction step, it seems that you need to know that the result I asked about is true. Since fiber bundles had just been introduced in the text, I thought there may be a short, elementary proof that the authors had taken for granted.
Here is an easy proof of the fact that a smooth fiber bundle over a contractible base is trivial. Bott and Tu's book, mentioned in the original question, work in this framework of smooth manifolds and maps. The proof uses Moser's trick and relies on the following general basic result.
Theorem: Every smooth fiber bundle admits a complete Ehresmann connection.
This was proposed as an exercise without proof in [Greub, Halperin, Vanstone: Connections, Curvature, and Cohomology I, Academic Press (1972)], it was presented as a theorem in [Michor: Topics in Differential Geometry, (GSM 93), American Mathematical Society (2008)] and other references with a gap in the proof, and it was finally established in del Hoyo: Complete connections on fiber bundles. Indag. Math. 27, 985–990 (2016).
Let $p:E\to B$ be a smooth fiber bundle, and let $h:f_0\cong f_1:B'\times I\to B$ be a smooth homotopy. Let us pick a complete Ehresmann connection for the pullback bundle $h^*E\to B'\times I$. Let $X\in\mathfrak{X}(h^*E)$ be the horizontal lift of the vector field $(0,\frac{d}{dt})$. Then the time 1 flow $\phi^X_1$ yields an isomorphism between $f_0^*E=h^*E|_{B'\times 0}$ and $f_1^*E=h^*E|_{B'\times 0}$. In particular, if $f_0=\rm{id}_{B}$ and $f_1$ is constant then $f_1^*E$ is trivial and so does $E=f_0^*E$.