Why are the transformer windings in series with line and load in this circuit?
That's not used as a transformer, it is referred to as a "common-mode choke".

It and the capacitors connected to it are there to minimize the amount of interference that conducts from the device into the power line from which it can radiate and interfere with other equipment. It also reduces the chance of any interference on the power line getting into the device. Apart from being an inconvenience to users this is required by the regulatory organizations such as the FCC.

By connecting the windings as shown the magnetic flux created by the current flowing in one winding is cancelled by the other so there is normally no flux caused by the current feeding the device - this allows the core to be smaller than otherwise.

This way of connecting the windings does not however prevent it offering a high inductance for common mode signals that can either cause interference or be caused by interference.

These common mode chokes are common in electronic devices, especially those with switching power supplies that tend to create a lot of interference.