It's hard to find credible details on these issues as they are a general political hot potato. This answer is based on my experience as a Belgian (Flemishman) who has worked in Brussels and has friends from the region.
neighbouring municipalities which are in Flanders but close to Brussels are undergoing francisation,
This francisation refers to native French speakers (implied to not speak sufficient Dutch) moving across the border into Flanders.
However, this does not change Flemish legislation regarding the administration of a language region. Flemish municipalities do their administration in Dutch, and if a person is not able to speak sufficient Dutch, it is up to them to cover for it (by e.g. hiring an official translator).
There are a few exceptions here ("faciliteitengemeenten") where both languages are provided for but these border towns are explicitly listed/required to do so.
This isn't just related to French speakers. My wife is a native English-only speaker and does not master Dutch well enough yet. For our wedding, we needed to hire a translator to ensure that she was able to be informed when signing the documents.
leading to conflicts between Dutch and French speaking residents (including edit wars on Wikipedia)
Those conflicts are nothing new. I don't intend to just dismiss your claim but the reality of it is that citizen conflicts occur, whether it's related to language, culture, religion, noise pollution, or any other grounds worth filing a complaint over.
Wikipedia is a relatively new phenomenon but it's simply another manifestation of the same problem. Here too the issues are not centered around language. There are edit wars over many topics (language, culture, religion, philosophy, opinion).
Additionally, the digital nature of the internet points towards this occurring regarding of whether the two "warring" factions live in the same town or not.
I think you're seeing a coincidence, where francisation of Flemish border areas and the popularity of Wikipedia (and subsequent edit wars) arose around the same time but are not necessarily related.
but the article does not state any specifics of how the Flemish government attempts to do so.
Cynical as it sounds, politicians promise a lot more than they achieve. This is no different. People easily get riled up about "those foreign language speakers who are coming in and changing our culture!" and politicians gain quick points by promising to tackle it.
But the linguistic/cultural divide is much harder to solve than a political promise makes it seems, so there aren't as many results as the initial promises would have you believe.
What specific measures does the Flemish government undertake to counteract the francisation of the Flemish rim/Brussels periphery?
If you're talking about what language people speak in their own home, the answer is none. The sole interest of the government in regards to francisation is with interaction between a citizen and their government.
Does Flanders cull the influx of French speakers into its region? No. In fact, Belgium has little to no border control in this regard: the European borders (at least the Schengen area) are open and people are free to move around; Walloons/Bruxellois are no different.
However, Flanders does enforce usage of its (only) official language, meaning it does not cater to multilingual documents in regards to administration between the citizen and their municipality.
Any further steps such as a mandatory language course for new residents are scoped to local municipalities and are not stipulated on a Flemish/federal level. Your question (as phrased) focuses on what the Flemish government undertakes.
From experience, reality often diverges from what is officially stipulated in these municipalities. As a straightforward example: a French speaking friend of mine moved to a municipality with mandatory Dutch courses, but was able to not have to go by simply signing a document that he agrees to using Dutch on official documents.