The primary output of the module is to switch on the HVAC system fan. As described before, in a normal thermostat, this is handled with a mechanical switch, a tilt switch, or with a relay output from an electronic thermostat. This module will work in parallel to the existing thermostat.
The most robust and reliable way to interface the module to the system is to use a relay, with relay contacts wired connected to the fan control circuit. This is much more tolerant of various voltage and current conditions and provides isolation between the circuits, but it also is the option draws the most current. For example, a small 3V relay that I have on hand draws 40mA. As noted in the prior log, this is no issue if the C wire is available, but it usually is not. Running off a pair of AA batteries, for 6 minutes every hour, the batteries would last in the range of 25 days, which is poor performance. Some digging suggests that there are some reed relays on the market which draw 10mA(5V), and I found one which draws 6mA(3V), so with some careful part selection this may be acceptable.
Another way is to use a TRIAC to switch. This does not provide isolation, and has more failure cases, but minimizes the current requirements. Specifically, current or voltage spikes are more likely to damage the part, and it will not work in a system where a DC control voltage may be used.