Shuck the universal garage door opener, pulling out the bare circuit board
Use a soldering iron to melt the leads for button #1 just enough that the button can be pulled out - you could also reflow the solder and pull out the button.
Replace the leads to the button with breadboard pins. For my remote I needed to wire up lead #1 and any other lead to connect the circuit - your milage may vary.
Wire together the exposed leads from the universal remote to a mini-breadboard, and complete the circuit with a pull-down resistor and a MOSFET as detailed in the circuit diagram. Connect the ground (not button #1) lead from the garage door opener to the source pin of the MOSFET, and connect the lead (button #1) pin to the drain pin of the MOSFET.
Connect a GPIO pin (I chose GPIO 17) from the Raspberry Pi to the gate pin of the MOSFET.
Connect a ground pin (I chose the ground pin next to GPIO 17) to the drain pin of the MOSFET. Bridge the 4.7k pull-down resistor between the gate and drain pins of the MOSFET.
You can optionally install the Honeywell HIH6130 I2C breakout to monitor temperature and humidity. This connects over the standard I2C pins on the Raspberry Pi; I used 3V of power however 5V can be used.
Follow the software installation instructions at https://github.com/deckerego/GarageSecurity/README.md