They pay game studios, Microsoft is the publisher.
Nintendo does the same for many of their games as well. Smash is developed by Bandai Namco and Sora Lts, Paper Mario is developed by Intelligent Systems, Pokémon is developed by Game Freak. These companies have various connections with Nintendo, like Microsoft have with their publishing game studios.
Nintendo does develop their core IP in-house though.
Halo was for the most part handled in house by MS owned studios, either Bungie or 343. Not really any different than how Nintendo handles Metroid Prime or Mario.
Nintendo seems to be a better guiding hand when working with external studios. Look at Silicon Knights output when partnering with Nintendo vs. MS for example. Or well, Rare for that matter.
Halo franchise is a good example of Microsoft culture wrecking a game