@heybusti, thanks for reaching out and for your interest in Rachio! You’ll probably need to install a pump relay (which the Rachio controller will trigger on/off) to power your pump.
For more details on pump relays, please see this support article. Additionally, here’s a video on how to install a pump start relay.
Do you know if you have any valves? From the photos provided, it appears your Intermatic is wired directly to the pump. Assuming so, I believe the Hunter PSR-22 will handle your needs and is available for $60ish on Amazon.
Hope this helps! Let us know if you have any questions.