Does it run cordova build android
then install the APK onto the device / emulator?
If not, what does it really do? I've tried searching for more details on cordova documentation site but to no avail.
Would appreciate any help
Yes, you're pretty much right. Citing from the Cordova CLI reference:
The run command ...
Prepares, builds, and deploys app on specified platform devices/emulators. If a device is connected it will be used, unless an eligible emulator is already running.