Jouer la carte de la facilité is a metaphor that means "to choose the least effort solution", i.e. avoid working hard to complete some task. There are several similar expressions based on card games, like jouer cartes sur table (play openly, to come clean), jouer sa dernière carte (last resort/chance attempt), brouiller les cartes (obfuscate/muddy the waters.)
La carte de la facilité is almost always used in a negative way, i.e. not when using the least effort is the smarter and recommended way but when the lack of effort is detrimental to the expected result.
Jouer la carte [de] can be followed by almost anything. Whether the whole expression is negative (the rarest cases), neutral (informative) or positive depends on what follows and on the context.
Here are various examples of cartes that can be played found on the Internet:
Jouer la carte de la séduction, de la prudence, de la transparence, de l'originalité, de la cohésion, de la sécurité, de la conciliation, de la solidarité, de la proximité, de l'offensive, de la qualité, de la régularité, de la gastronomie, de l'apaisement, de la démagogie, de la colère, de la traduction, de la politique de la peur, de l'union,…
In the 20 Minutes article, the article headline is more than likely a play of word on that expression as the content is not telling Microsoft is lazily choosing the least effort way but on the opposite that they substantively work to improve their product and provide an easy to use solution to their customers.
Headlines are here to attract the attention, they are une accroche, a bait to hook the reader. Should the title had been something like "Microsoft improves the user experience with Office 2016", some readers might have avoided to even start reading the article, fearing a boring content and perhaps suspecting some impartial advertising disguised as news.