How do I sniff this RF Transmitter Gate Opener Circuit?
I'd definitely recommend getting a BusPirate, you can probably find the frequency in the developer docs and then do something like this:

http://ms3c.wordpress.com/2011/06/19/basic-rf-sniffing-with-the-bus-pirate/

Failing that you can probe all the pins with the BusPirate logic analyser mode and just observe the transmit waveform.