I think the key is the little may be hidden in there (different paragraph from the one you quoted). If you leave "scanning always available" on then basically the wi-fi will blip on and off every so often just to see what's around... this could be enough for a nearby wireless network to detect/track you. If you read the google plus post (and bare in mind it's by the person who wants you to download his software) it actually points out you can still turn your wi-fi off in latest Android (off in the sense you'd expect - "One solution is shutting off wi-fi completely").
The paragraph you quoted is a bit of a dogs breakfast... the first part happens (wi-fi networks will be able to see the MAC address of any device polling them). The only reason the second part would happen is when your device knows about wireless networks with
SSID broadcast turned off (it used to be recommended to turn this off, but now the opposite is true for this reason) - if it knows of such cloaked networks it'll poll each network it meets to see if it's one of the ones it knows (simplified). There's also allegedly some "known network sharing" between your device and google for the purpose of backing up your network settings to the cloud, but this isn't available/visible to the wi-fi network.
So the benefits... disable wi-fi completely a strange feature since the author recognizes you can do this with stock, but you wouldn't want to because you'd loose the "scanning always available setting". Random MAC address this could be useful, it certainly defeats a series of networks from tracking your device's movement, but so does turning of wi-fi completely. MAC addresses are supposed to be unique, and while it's a large namespace, one wonders with a randomizer if collisions will happen (if two devices have the same MAC address they won't work properly on wi-fi). If you visit places that have MAC address filtering or identification (for legitimate reasons - your home network, a business network, a company that provides free wi-fi for a period and tracks that you've accepted their terms) you may find your device no longer works although there's mention of a setting that will restore your stock MAC when wi-fi is on. war mode sounds like it just changes the MAC address more frequently (possibly during a wi-fi session) but this isn't without processing overhead.