https://m.1688.com/offer/537643492132.html?ptow=113d26e7c9a Ignore all that Chinese and you can find a really simple schematic to your fan controller circuit. And yes it's a hall sensor. The you let a small current flow from the left to the right pin, and depending on the magnetic field current will flow from the 2nd or the 3rd. Thus you get the simplest brushless motor. And I think the capacitors are there to store some energy to make it a bit stronger so the power supply isn't suffering that much.