Rain sensor can operate in one of two ways. It can either act as a sensor input with more modern controllers, or it can physically disconnect the common wire (useful for controllers which do not have sensor input).
In your case the rain sensor was setup to interrupt the common, even though the SEN input was available. I guess whoever was setting this up was familiar with older, dumb, controllers and set up the rain sensor in the old fashioned way.
In your Rachio setup, you should connect a thicker white cable, which you see coming out of the blue twist connector, to one of the C terminals. Thick white cable is your "true" common wire.
You can also continue to use the rain sensor if you connect the thin red wires to the SC and SP terminals and connect thin brown wire, which is currently connected to C terminal, to SC terminal (yes, you will have two cables connected to one terminal, it may be a good idea to twist them together before inserting them as one) while the thin white wire would connect to S1 terminal. Do not forget to turn on the Rain sensor option for S1 input within the software to make use of your rain sensor.