I'm also curious if the mechanics of that shift were unique to Belgium, or if it's something that's been repeated in other European cities as well (though I understand that most countries don't have Belgium's particular language politics).
This is an interesting question and as someone from Brussels I might be able to answer this.
In Belgium (and even Flanders), a lot of upper class people spoke French since the country was first created and explains why a lot of the information could only be found in French at first (courts, administration, army, media and (most importantly) schools.
Around the 1880's, there was an uprising in the number of Flemish people understanding French and speaking it and this would continue to grow in the next decades.
This makes it so that after some time (date seems to be around 1910) a lot of Flemish citizens of Brussels start to avoid passing Flemish to their children in favor of French. Added to that is the fact that a lot of new people came in Brussels who mostly spoke French.
In the late 1950's, early 1960's new migrations to Brussels started from countries that mostly spoke French which added once more the the reason Brussels became French.
Lastly, in the late 20th century, when Brussels became more and more an international hub (European commission, NATO, companies, ...) wich lead to French again gaining in popularity and even more languages coming in than the oringal Dutch and French.
One last point, and I'll just put it here as this is more a personal opinion, as I was growing up in the late 80's in Brussels, I saw a lot of Flemish people from the region move out of it to get bigger, more green and possibly safer housing outside of Brussels. This might have been one of the final straws on the Flemish in Brussels as I remember a lot of people speaking Flemish in our neighbourhood in the early 80's and by the end of the 90's almost no one.
TL;DR: Immigration + flemish people stopping to speak Flemish is the reason.