What should I do with exposed wires from a yard irrigation manifold?
It looks like the Green or Green/White wire is also hooked up. He used 4 twisted pairs, most commonly used as Ethernet wire (the one that you use for internet on your computer, router, modem, etc.) and is most likely a Category 5 cable. It is a relative cheap cable that has 8 wires, looks like he only needed a few but it is easier and cheaper to use what you have sometimes.

As long as your system works and the wires that are connected aren't getting hot (meaning the wires are too small for the electric load going through them) then you're fine. The connectors to the attached wires look like waterproof ones and sounds like you can trust the contractor on that one. 

I'd wait a while to make sure everything works, or ask them to do it if they're coming back already, and cut the unused wires close to/at the sheathing (blue covering in your example) and put some tape around that to ensure you don't trap some water in there. 

The unused/exposed wires left alone can/will have some copper corrosion form but they are unused so it's more of an aesthetic concern which you can address.