$\newcommand{\C}{\mathbb C}
\newcommand{\CP}{\mathbb{CP}}
\newcommand{\GL}{\mathrm{GL}}
\newcommand{\Z}{\mathbb Z}
\newcommand{\PGL}{\mathrm{PGL}}$
Here's a topological proof.

An algebraic $\CP^k$-bundle $X\to\CP^n$ is a fiber bundle with structure group $\PGL_{k+1}(\C)$. It therefore is
equivalent data to a principal $\PGL_{k+1}(\C)$-bundle $P\to\CP^n$: given $X$, the fiber of $P$ at $y\in\CP^n$ is
the $\PGL_{k+1}(\C)$-torsor of isomorphisms $\CP^k\overset\cong\to X_y$. Conversely, we can recover $X$ from $P$ as the
associated bundle
$$X = P\times_{\PGL_{k+1}(\C)} \CP^k := P\times\CP^k /\{(p\cdot g, q)\sim (p, g\cdot q)\mid g\in\PGL_{k+1}(\C)\},$$
with the map to the base induced from that of $P$.

$X$ extends to a complex vector bundle iff we can lift $P\to\CP^n$ to a principal $\GL_{k+1}(\C)$-bundle
$P'\to\CP^n$; the complex vector bundle in question is the associated bundle
$$P'\times_{\GL_{k+1}(\C)}\C^{k+1}\to\CP^n.$$
The isomorphism class of $P$ is equivalent data to a homotopy class of maps $f_P\colon \CP^n\to B\PGL_{k+1}(\C)$,
where $BG$ denotes the classifying space of $G$; lifting $P$ to a principal $\GL_{k+1}(\C)$-bundle is equivalent to
finding a map $f_{P'}\colon\CP^n\to B\GL_{k+1}(\C)$ whose composition with the map $\phi\colon B\GL_{k+1}(\C)\to
B\PGL_{k+1}(\C)$ recovers $f_P$ up to homotopy.

We can use obstruction theory to prove that a lift exists, since all of these spaces are CW complexes. Suppose we
have a lift on the $m$-skeleton of $\CP^n$; then the obstruction to it extending to a lift on the $(m+1)$-skeleton
is an element of $H^{m+1}(\CP^n;\pi_m(F))$, where $F$ is the fiber of $\phi$, which is $B\C^\times\simeq K(\Z, 2)$.
It therefore suffices to show that a lift exists on the $2$-skeleton, as $\pi_m(K(\Z, 2)) = 0$ for $m\ne 2$.

In the standard CW structure on $\CP^n$, the $2$-skeleton is $\CP^1\cong S^2$, so we want to lift from $[S^2,
B\PGL_{k+1}(\C)] = \pi_2(B\PGL_{k+1}(\C))$ to $[S^2, B\GL_{k+1}(\C)] = \pi_2(B\GL_{k+1}(\C))$. Associated to the fibration $F\to
B\GL_{k+1}(\C)\to B\PGL_{k+1}(\C)$ we have a long exact sequence of homotopy groups
$$\dots\to\pi_2(B\GL_{k+1}(\C))\overset{\phi_*}{\to}\pi_2(B\PGL_{k+1}(\C))\to \pi_1(F)\to\dots$$
but $\pi_1(F) = \pi_1(K(\Z, 2)) = 0$, so $\phi_*$ is surjective, and a lift exists.