The first step is to find out how your garage door opener works. What you need to figure out is what happens when you press the physical switch (assuming you have one). If all you have is wireless remotes then you have to look closely at the garage door motor housing and see if there is a pair of contacts that can be used (shorted?) to trigger the "close/open the door" action. Once you have that understood, you can proceed to try and build this.

I can't say I am an expert in all types of garage door opener motors. I have a Chamberlain system and the physical button next to the door of our house just has two wires that run to a pair of contacts on the door motor. When these are shorted, the door goes down or up (toggles state).

If your door is different, you may need to figure out another approach. But basically you are looking for a way to electronically trigger the door state change.

Once you know how the garage door is electronically triggered, you need to figure out if you can use either a Darlington NPN transistor or maybe just a relay to trigger the door. One of these must be used as the Arduino itself is not capable of doing it directly in most cases. Why? Because the voltage and possibly current demands are outside of what one of the digital I/O pins is capable of.

Buy a WeMo and take it apart and grab the low voltage board.

Next I would suggest hooking up the Reed Switches (open and closed sensors)  and WeMo to the Arduino board and loading up the code and testing the logic. Feel free to add some serial.println commands to view the state of the FSM, compare that to the Arduino's built in LED. Testing and understanding how it all works will make the installation a whole lot easier.

To install, mount the Reed switches to the frame of the garage door and one of the magnets (that comes with the reed switches) to the door itself such that when the door is open it trips one switch and closed trips the other. Use a multimeter to measure resistance on each Reed Switch to know they are tripping! And there are two sets of contacts on each Reed Switch (Normally Open and Normally Closed). You can use either but you will need to change the Arduino code if you do not use the normally open contacts (as I did).

Before hooking the Garage Door motor to the Arduino (so that the Arduino can talk to the garage door and tell it to move), test the door extensively by causing it to change states with one of your old car remotes, the WeMo App on your phone, etc. Verify that the WeMo stays in sync with the door state. So if the WeMo is ON (blue LED is on), the door is up and visa versa. Also verify that the Arduino's state machine is also in sync such that the orange LED is ON meaning the garage door is up and the WeMo light should be ON. Or that the Arduino LED is OFF meaning the door is down and the WeMo light OFF. And lastly, that the arduino LED is flashing while the door is moving/in transition/or stuck in between (some doors can stop part way).

If things are working well at this point, might be time to try hooking the garage door motor up to the Darlington NPN Collector and Emitter as shown in the diagram. You do have to be sure your garage door has a +15V and 0V contacts that the Darlington NPN will short together properly. My door works this way. Yours may not! I can't be responsible for a door that works differently and which you damage because of your lack of understanding of electronics.