Widest path algorithm steps
The way to use Dijkstra's algorithm to solve the $s$-to-$t$ widest-path problem (in a directed graph) is described well in the following lecture notes:

http://www.cs.cmu.edu/~avrim/451f08/lectures/lect1007.pdf

Make sure to read Sections 13.1, 13.2, and 13.3.  You will see that a single-line change to Dijkstra's algorithm suffices (we change the $+$ to $\min$, and change the $\min$ to $\max$, to represent the fact that we now have a different metric for the "goodness" of a path).

The section you quoted from Wikipedia is talking about a small optimization to this variant of Dijkstra's algorithm.  However, you can use the variant of Dijkstra's algorithm in Section 13.3 of those lecture notes directly, without the optimization mentioned in Wikipedia, if you like -- and that's easier to understand.