The following is an excerpt from the Atiyah's K-Theory.
If $E$ is a vector bundle over $X$ then each point $a\in P(E)_{x}=P(E_{x})$ represents a one-dimensional subspace $H_{x}^{*}\subset E_{x}$. The union of all these defines a subspace $H^{*}\subset p^{*}E$ where $p:P(E)\rightarrow X$ is the projection. It is easy to check that $H^{*}$ is a sub-bundle of $p^{*}E$. In fact, the problem being local we may assume $E$ is a product and then we are reduced to a special case of the Grassmannian.
In the statements above, $P(E)$ denotes the projective bundle associated to $E$. I am having some difficulty with the last statement about the Grassmannian.
(Small caution: Algebraic geometers call the bundle $H^*$ the tautological bundle; the canonical bundle refers to the top exterior power of the cotangent bundle. When I was a student, topologists were coming around to this convention as well.)
Presumably Atiyah means that to understand the tautological bundle of a projective bundle $\mathbf{P}(E)$, it's enough (locally) to understand the tautological line bundle over a projective space (a.k.a., Grassmannian of lines).