What values are you injecting to manually change the output state?
If you will eventually want to apply hysteresis then I suggest you look at node-red-contrib-ramp-thermostat which will provide the comparison with hysteresis for you. You can ignore the ramp component of the node.
Alternatively if you are trying to achieve temperature control then have a look at node-red-contrib-pid which will allow you to perform pid control which can be a lot better than thermostatic control.
However, if you just want to generate true or false (or 1 or 0 or whatever) based on a simple comparison then you can do it in a change node like this.
Finally, I suspect the problem with your flow is that you are setting the payload to the strings "true" and "false" rather than boolean true and false. In the Change nodes click the az dropdown and change them to boolean.
