The pump start relay and Rachio combo is what is going to replace the current intermatic timer you have now. The pump start relay is going to handle the heavy lifting of the 120v pump wiring while taking a low voltage 24Vac nudge from Rachio to fire the relay, thus running the pump. Without knowing the exact model of pump start relay you have, this is what I would ASSume (so don’t make an ass out of me!)
L1in and L2in would be wired to the power in side of the current Intermatic timer. L1out and L2out will connect to the “to pump” side of your Intermatic timer. The 24Vac lines will become the common and Master Valve/Pump for the Rachio. These really shouldn’t matter how they are wired since polarity doesn’t matter. Grab some 18 gauge wire and run one leg of the 24V to the C terminal of Rachio, and the other leg to the M terminal of Rachio.
Once you get the wiring done, it will be a pretty straight forward setup of Rachio. Make sure you specify that you have a master valve/pump. Like I mentioned earlier, you will need to create a zone in Rachio so you can create a watering schedule. This will be a ghost zone since all we are trying to do is fire the master valve/pump when the “zone” runs.