The girth or cinch is the strap that secures the saddle to the horse's back.
So why do some western saddles have two of them?
You often see a western saddle with both a primary cinch and a rear or back cinch.
The back cinch was invented for roping. When you rope from horse back, you secure the rope to the horn. This pulls the front of the saddle forward and down when the cow protests. The back cinch prevents the saddle from tipping too far forward.
Back cinches are also seen when doing serious trail riding, such as mountain work, to stop the saddle (and rider) from tipping forward when going down steep hills. A lot of barrel racers use them (because you need everything to keep the saddle stable during those turns).
Many cowboys will also put a back cinch on a horse that is prone to bucking, as it can make it easier to stay on.
Other people prefer to avoid the use of one as much as possible - if not correctly fitted they can annoy the horse into bucking, and some horses absolutely hate them. To keep the rear cinch from slipping back, a "cinch hobble" is used to secure the rear cinch to the main cinch.
So, that's what the second girth is for.
(You can clearly see the rear cinch on the roan horse).