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.
Ground-fault circuit interruption (GFCI) receptacles, are not wired the same as regular duplex receptacles. In a standard duplex receptacle, both receptacles and all terminals are directly connected together (Unless modified). If one half of the receptacle is powered, then the other half is as well. With a GFCI receptacle there are LINE side terminals and LOAD side terminals, which are separated by an internal switching mechanism.
The wires feeding the circuit are connected to the LINE terminals, which supplies power to the device. If everything is wired correctly, there are not ground-faults, and the device is not tripped, then electricity is allowed to flow to the receptacles on the device and to the LOAD terminals. So if everything is functioning as it should, there should be be power at both the LINE and LOAD terminals. However, if the GFCI device is tripped, there will only be power at the LINE terminals.
Resetting the GFCI
If the GFCI has tripped, it can usually be reset simply by pressing the RESET button. If you press the RESET button and don't feel/hear a click and/or the button doesn't stay in, it means there is a problem and the internal mechanism is not allowing the GFCI to be reset. You can try pressing the TEST button, then pressing the RESET button again making sure you press the RESET button all the way in. If the device still will not reset, you'll have to try and determine the reason.
Why won't a GFCI device reset?
There are three reasons a GFCI device will not reset.
Wiring is wrong
If the GFCI device is not wired properly (LINE and LOAD reversed, hot and neutral reversed, etc.), the device will not allow a reset.
There is a ground-fault
Obviously, if there is a ground fault, the device will trip as soon as you try to reset it.
There is a problem with the device
If the device has gone bad, it will (should) not reset. Some devices will continue to hold, even if there is something wrong internally. However once they trip, they cannot be reset. Other devices will trip as soon as something internal dies, and will not reset. This is why monthly testing is suggested. If you press the TEST button, and then are unable to reset the device. You'll be made aware of a problem sooner, and can have it repaired (hopefully) before any damage is done.
Rewiring a new device
Before you begin, turn off the power at the fuse/breaker box and make sure it's off.
Locate the supply wire pair
There should be an ungrounded (hot) and grounded (neutral) conductor pair (likely as part of a cable assembly), that supplies power to the circuit. As it sounds like you've already located these, I won't go into detail as to how to locate them here (there are many other answers on the site that explain this procedure).
Terminate the supply wire pair
- Connect the bare/green grounding conductor to the green grounding screw on the receptacle (and to the box if required), and to any other bare/green grounding conductors.
- Connect the (white) grounded (neutral) conductor to the silver colored screw terminal labeled LINE on the device.
- Connect the (black) ungrounded (hot) conductor to the brass colored screw terminal labeled LINE on the device.
Terminate load side wires
If there are devices downstream that require GFCI protection, you'll have to connect the wires feeding those devices to the LOAD side terminals on the device.
- Connect the bare/green grounding conductor to the grounding conductors in the box.
- Connect the (white) grounded (neutral) conductor to the silver screw terminal labeled LOAD on the device.
- Connect the (black) ungrounded (hot) conductor to the brass screw terminal labeled LOAD on the device.
Set the device
Once all the wires are connected, install the device in the box using the mounting screws. Install the face plate, and turn the power back on.
- Press the RESET button.
- You should have power to both receptacles, and any downstream devices.
- Press the TEST button.
- You should no longer have power at the receptacles, or any downstream devices.
- Press the RESET button again.
- Power should be restored to the receptacles, and downstream devices.
Best Answer
It's a given that any wires in cordage will be stranded. Being designed for a different kind of service is what makes cordage different from in-wall wiring, and why you should not use one for the other.
So it's a given that any plug whose primary use is cordage is designed to work with stranded wire. Here, you should follow the instructions and labeling on the plugs, because following those instructions are the basis for the testing and certification which the plug received from your NRTL. (Nationally Recognized Testing Laboratory) such as UL, CSA, TUV etc.
Not following the instructions is an "off-label use" (to put it kindly) and can result in an illegal install and non-payout of fire insurance.
Speaking of non-payout, you need to be careful what you are sticking on mains power, and in particular, built into walls. Random electronic components out of the Digi-Key catalog are unfit for use in mains power. Equipment made for use in mains wiring can get its cert/listing faster if it is built with listed components, that is all a component listing means, it does not mean you can use components directly.
So for instance "random relay" is not listed for use in mains, but an "RIB" unit containing that same relay can be used. (RIB ~= Relay In Box).
If your aim is smart appliance control, you may be better off using existing, listed/certified smart devices (i.e. Not Sonoff), and having your electronics communicate with them. Generally, the rules are greatly relaxed for homebrew equipment when it is entirely low voltage and fed by a listed/certified wall wart.