Oh i get your question, I got one of those:
http://www.aliexpress.com/snapshot/311636125.html
It is controled by serial port, so the tlg is hook on its serial port directly
But the tlg has a gpio that can be turn on and off to control a relay, i use it in a rfid project. On the picture you have just posted, the pin G13. Is the gpio pin. It can be turn on and off by Arduino using AT command or by external system using telnet and AT command.
Yeap 300 mA under 3.3v. So nope the 3.3v with 50 mA of Arduino is not enogh. You need an external power source for sure.
Oh by the way, it's Chinese New Year. So if you got the module from freezone, dont expect reply from Jane until tomorow since she wont be back until the 5 feb China time.