The signal current from the transformer flows in one input pin of the diff amp and flows out of the other input pin in the diff-amp so the current in the secondary circulates in a loop with the two input pins on the diff-amp.
Since the secondary is floating it's common mode voltage will float around and automatically adjusts itself to be within whatever range the diff amp wants it to be (since the diff-amp is connected to ground and this somehow relates back to the inputs inside the op-amp circuitry). This is ONLY able to happen because the secondary is floating.
If the secondary was something that was not floating (i.e. if it was another type of device) then it would mean it had it's own ground voltage, separate from the diff-amp. In that case, the source has it's own ground and the diff-amp has it's own ground and if they are not connected then the two grounds voltages (measured with respect to the surroundings...probably the earth itself) become different from each other and never adjust to each other. The grounds could become different enough that the common mode voltage on the two source outputs got too high for the diff-amp to handle.
Do you understand? Each circuit has it's own "ground" which is 0V for everything in the circuit. But all voltages are relative so two circuits isolated from each other can have their own "0V ground" are in their own little universe. But that can cause problems if you try to connect two circuits together without connecting the grounds so that the grounds are the same. The grounds are measured with respect the surroundings (i.e. the earth itself). which is considered to be the "true 0V".
I guess each circuit is like a different planet and each one has it's own buildings (voltages and circuits) sitting on the surface of each planet (ground). But if you try to build a bridge between two buildings on two different planets, it's a problem unless you somehow ensure that the surface of the planets do not move with respect to each other or else your bridge will break apart (i guess connecting the grounds in two circuits together is a bit like...bolting the surface of two planets together?).