I came across this switching power supply circuit and I noticed the transformer is connected in a way I had never seen before. What is the benefit of connecting it like that? And why does the circuit use a capacitor voltage divider?
In order to understand the circuit, you need to understand what it is trying to accomplish. Primarily, it is a high frequency (low pass) line filter. This is accomplished with a capacitor and an inductor, for each line (4 components).
Some smart person figured out that the windings of a 1:1 transformer could be used as the inductor with the advantage of reducing the component count (3 now) and the additional benefit of cancelling out any common mode "noise" on the lines. It is this last benefit, that requires the use of a "transformer," to keep each "inductor's" characteristics, as equal as possible (same core, materials,number of turns, etc.).
In conclusion, the answer is, that each transformer winding is being used as an inductor, that's why they are connected as shown.
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.