I agree this isn't completely obvious. Here's a slightly different take on it. Our intended vector bundle is
$E/E' :=\coprod E_x/E'_x$,
the disjoint union of the quotient vector spaces of fibres. We just need to specify the topology on it. We do this by describing a family of maps which we intend to be continuous local trivializations for the bundle.
So, take a point $p\in B$, and a nhd $U$ of $p$ on which we have a frame $(e_1, \cdots e_{n+k})$ for $E$. Choose $1\leq i_1<\ldots< i_k\leq n+k$ such that on the fibre $p$,
$E_p=E_p' + span(e_{i_1},\ldots e_{i_k})$.
By the continuity of the determinant function, in fact there's a neighbourhood of $p$ on which this is true; that is, there's a (perhaps smaller) nhd $V$ of $p$ such that for all $x\in V$,
$E_x=E_x' + span(e_{i_1},\ldots e_{i_k})$.
So at each $x\in V$, we have a basis
$(e_{i_1}+E_x',\ \ldots \ e_{i_k}+E_x')$
for $E_x/E'_x$. We demand that this collection of bases give a (continuous) frame for $E/E'$ over $V$. It's an easy check that the transition functions between two thus-constructed local trivializations are continuous, as required.