These aren't necessarily my opinions, but Microsoft might be thinking along the following lines...
Free Windows everywhere might firstly (in Microsoft's eyes) mean bye-bye SteamOS, keeping Linux in its all-too-niche position in the games market (and if both OSs are free, why would someone choose an OS with far less of a games catalogue other than if they're running a system that really needs a less resource-heavy OS). It would also ensure MacOS remains an unsuitable choice for mainstream gaming. If Windows holds the greatest market-share it stops other platforms gaining a secure foothold, then games and software will continue to need Windows, which in turn secures Microsoft's future.
Free Windows will knock a fifth off the price of a decent budget gaming PC, placing it firmly with the latest-gen consoles on pricing. This will firstly help to keep the Steambox at bay, and will arguably make choosing a PC as a gaming platform more attractive given the much bigger and diverse games catalogue (especially given the number of PCs with small form factors and their ability to connect direct to TV via HDMI, as well as stream games all around the home). For PC gamers who also want a console, a free Windows OS that can share games with the XBOne secures a better future for the perhaps less-loved XBOne.
The situation Microsoft WON'T want to have happen is a rash of games made as SteamOS exclusives. It could be that Valve might just have the financial clout to sponsor this whereas Sony (for Playstation) will not because the PS4, like the XBOne, is sold at a loss with profits coming from game platform fees. Valve takes a whopping 30% fee from every title sold on Steam so maybe it would be rather easy for them to offer some quality developers a 30% bonus for exclusivity. Microsoft is clawing back as much platform support as they can given the debacle that is Windows 8.1 and they now seem to be reneging their argument that gaming is shifting to tablet and shifting it back firmly to PCs and consoles (their console, that is).