The most easy and straightforward way is to just emulate the pushing of the button. That way you don't need to know anything about how the transmitter does it's work!
Meaning you use a (spare) transmitter for your alarm system and use a relay to act like the push button. You connect this relay (the contacts that get activated by the relay, common and normally open) parallel with the original push button on the transmitter. You connect the coil of the relay to a GPIO pin on the RPi using one of the many examples on this site or something like described here, and you're done with the hardware. Write a simple script or program that activates the GPIO pin/relay for a moment and you're done....
As to contact bouncing:
In an ideal world the relay closes the contact and that is it, as does the button on your remote. But in reality both contacts bounce (make and break contact repeatedly in a very short period of time). For normal buttons (manually operated) this problem is normally a bit severe and happens over a longer period of time, because your hand/finger does not push the button in a constant way everytime. So when your remote operates by hand in a stable manner, we can safely assume that there are hardware and/or software solutions to this problem already in the remote, thus the relay (which is faster and more constant in closing it's contacts every time) will not have a problem operating the remote in a stable way. If for some reason the alarm switches on and off in a very short period of time (while you know the relay contact did not change state), you experience contact bouncing and you need to address that. You can solve that in multiple ways, the easiest is just adding a capacitor and resistor (effectively making an RC filter), if that does not solve it, you can add a Schmitt trigger. But before overflowing you with scenarios that probably are not be needed (because there already must be a form of debouncing used in the remote), try the plain solution first (locally not debounced) and if the bouncing problem appears, open a question about it and you'll get all the info needed.
You can use this over SSH, you could make some 'server' application with a client for your cell phone and turn the alarm on and off from your cell phone, the sky is the limit.
Reproducing the signal from scratch is a bigger challenge ofcourse, but you need to have a lot of info about the system, or you need to spend a lot of time trying to figure out how that thing does what it does...