How do I remove wires from thermostat without screws holding the wires Or any apparent hole to push and loosen the wires. Picture below shows the wires.
That is very easy. Just push in on the little tabs that are in place of the screws while pulling on the wires until they release. As you can see, the sockets with wires attached have lower tabs than the ones with no wires. That is because of the pinch mechanism inside of the sockets that is set up so that when the tabs are depressed enough, it allows the wires to be moved in and out. And when released, they pinch down, on watever is in the socket; if there is a wire there, it will be gripped.