I don't know how to comment rather than answer, but Craig, the equivalence is really not mysterious: the zero section of the universal complex line bundle gives an equivalence from $\mathbb{C}P^{\infty}$ to the total space of the unit disk bundle $D$. The total space $S$ of the unit sphere bundle is really just $S^{\infty}$ (think of the universal principal circle bundle), which is contractible, and the quotient map $D\to D/S = MU(1)$ is an equivalence. The equivalence $\mathbb{C}P^{\infty}\to MU(1)$ is the composite of these two equivalences.
Let me add something a conceptual aside about orientations of fibrations. Let $p\colon X\to B$ be a spherical fibration with fiber $S^n$, such as the fiberwise $1$-point compactification of an $n$-plane bundle and let $E$ be a ring spectrum. In ``Parametrized homotopy theory'', Johann Sigurdsson and I show how to make sense of the parametrized spectrum $X\wedge E$ over $B$ with fibers $E$. An orientation (in the classical cohomological sense) is the same thing as a trivialization of this parametrized fibration, that is, an equivalence from it to the parametrized spectrum $(B\times S^n)\wedge E$ over $B$B$.
I should admit that my answer is digressive. Section 5 of my paper ``What are $E_{\infty}$-ring spectra'' relates universal $E$-orientations of $G$-bundles to maps of ring spectra $MG \to E$.
