It's a good idea to protype your circuit user a solderless breadboard before soldering it up. On Trinket, the positive red wires should go to USB+ for testing over USB power, then moved to BAT+ later for battery power.
Wire up your breadboard or soldered circuit according to the circuit diagram noting the above.
Use black gaffer tape to secure your laser to the flat paddle servo attachment and stick it on the servo gear. Do not screw it on yet, we'll adjust the position after testing the servo rotation.
You may need to solder longer wires on your laser diode for testing purposes.
Prepare the side of your dog goggles by making sure they're clean and dry.
Use hot glue to temporarily mount the servo to the goggles. This is a weak connection that we will use for testing/fitting purposes only, and is to be replaced with a more heavy duty adhesive in a later step.